First, remember that integrals can be broken up over addition/subtraction and multiplication by constants. In this process, we may have to apply the properties of integrals (like c f(x) dx = c f(x) dx). Since this is a transcendental function, specifically a trigonometric function raised to a numerical exponent, we can list down this form of power rule formula for our reference: Lets identify the transcendental function and the numerical exponent from the given problem: $$\frac{d}{dx} (u^n) = \frac{d}{dx} ((\sin{(x)})^2)$$, $$\frac{d}{dx} (u^n) = nu^{n-1} \cdot u$$, $$\frac{d}{dx} (u^n) = 2 \cdot (\sin{(x)})^{2-1} \cdot \cos{(x)}$$, Simplifying algebraically and applying applicable trigonometric identities, we have, $$\frac{d}{dx} (u^n) = 2 \sin{(x)} \cos{(x)}$$. (2x2 - 3x) dx = 2x2 dx - 3x dx ( (f(x) + g(x)) dx = f(x) dx + g(x) dx) Hence, $$\frac{d}{dx} (x^n) = \frac{d}{dx} (x^{-\frac{5}{2}})$$, $$\frac{d}{dx} (x^n) = \left(-\frac{5}{2} \right) \cdot x^{\left(-\frac{5}{2} \right)-1}$$, $$\frac{d}{dx} (x^n) = -\frac{5}{2} x^{-\frac{7}{2}}$$, $$\frac{d}{dx} (x^n) = -\frac{5}{2x^{\frac{7}{2}}}$$, $$f'(x) = -\frac{5}{2 \hspace{2.3 pt} \sqrt{x^7}}$$in radical form. We already know that the inverse process of differentiation is called integration. Therefore: \(\begin{align} \displaystyle\int 2x^3 + 4x^2 \text{ dx} &= \displaystyle\int 2x^3\text{ dx} + \displaystyle\int 4x^2 \text{ dx}\\ &= 2\displaystyle\int x^3\text{ dx} + 4\displaystyle\int x^2 \text{ dx}\end{align}\). Thus we take the exponent of the base and multiply it by the coefficient in front of the base. \[f(x) = a.x^{-n}\] For the constant, remember that the integral of a constant is just the constant multiplied by the variable. Finally, add C to the final result (the integration constant). This formula is illustrated wih some worked examples in Tutorial 3. For example, the integrals of x 2, x 1/2, x -2, etc can be found by using this rule. 3.1 The Power Rule. \end{aligned} \], We integrate \(\int 6.\sqrt{x} dx\) as follows: The integrand is the product of two function x and sin (x) and we try to use integration by parts in rule 6 as follows: Let f(x) = x , g'(x) = sin(x) and therefore g(x) = . See the pattern? Take a look at these pages: window['nitroAds'].createAd('sidebarTop', {"refreshLimit": 10, "refreshTime": 30, "renderVisibleOnly": false, "refreshVisibleOnly": true, "sizes": [["300", "250"], ["336", "280"], ["300", "600"], ["160", "600"]]}); $latex f'(x^{n_k} + + x^{n_2} + x^{n_1} + c) = {n_k} x^{{n_k}-1} + + {n_2} x^{{n_2}-1} + {n_1} x^{{n_1}-1} + 0$, $latex n_k$ = the exponent of the algebraic term with the highest degree of exponent in the polynomial, $latex n_{\texttt{\#}}$ = the exponents of the other algebraic terms in the polynomial, $latex c$ = constant, which if derived, is equal to zero, $latex u =$ the derivative method of the transcendental function, $latex n =$ the numerical exponent of the transcendental function. y 4 dx = y (4+1) / (4+1) = x 5 /5 Sum Rule of Integration Integration can be used to find areas, volumes, central points and many useful things. Experienced IB & IGCSE Mathematics Teacher The general strategy is always the same if you dont already have exponents, see if you can write the function using exponents. Power Rule for Derivatives Calculator. The power rule of integration is an important and fundamental formula in integral calculus. Then, apply the power rule and simplify. The power rule for integration is an essential step in learning integration, make sure to work through all of the exercises and to watch all of the tutorials. Rewrite using algebra before you apply calculus rules so that you can use the power rule. One more old algebra rule will let us use the power rule to find even more integrals. & = 2x+c \end{aligned}\], We integrate \(10x^4\) as follows: The power rule of integration can be written in terms of any variable as exampled here. Using this fact we can integrate any function written as: & = \frac{2}{1}x^1+c \\ Useful Trick: it's often useful to use the fact that \(\int ax^n dx = a \int x^n dx\), particularly when \(a\) is a fraction like in question 5. in the following exercise. The two parts are correlated. Given a function, which can be written as a power of \(x\), we can integrate it using the power rule for integration: It is x n = nx n-1. The derivative of a function is 6x 2. These are the few elementary standard integrals that are fundamental to integration Constant Rule If we have any constant inside the integral then it is to be taken outside. \end{aligned}\], We integrate \( \int \sqrt{x} dx \) as follows: Did you notice that most of the work was with algebra? In this case, our exponent is 12. We can integrate polynomials, negative exponents, and, 'n' is any real number other than -1 (i.e., 'n' can be a positive integer, a negative integer, a. Functions looking like \(f(x) = a.\sqrt[n]{x^m}\) can be written as powers of \(x\) using fractional exponents: Integration is simply the inverse process of differentiation.We use integration in mathematics to find areas, volumes, etc. However, in cases where other function is inverse trigonometric function or logarithmic function, then we take them as the first function. Using the power rule, observe that \int (5x^4) \,dx = x^5 + C = F (x) (5x4)dx = x5 + C = F (x). Examples of the power rule in effect are shown below: x 6 = 6x 5 x 8 = 8x 7 x 3 = 3x 2 x 8 . By doing this, we will have a single variable raised to a negative rational numerical exponent. Learn the why behind math with our certified experts, Integrating Negative Exponents Using Power Rule, Applications of Power Rule of Integration, Radical functions (like x, x, etc) as they can be written as exponents, Some type of rational functions that can be written in the exponent form (like 1/x, The integral of any constant with respect to x is the. & = - \frac{1}{x^2} + c \end{aligned}\], To integrate \(\frac{3}{x^5}\) we use the fact that \(\frac{3}{x^5} = 3x^{-5}\): Hence, the power rule is applied to integrate polynomial functions. i.e., the power rule of integration rule can be applied for: The power rule says that: xn dx = (xn+1) / (n+1) + C (where n -1). Example: Integrate $$\left( {\sqrt[3]{x} + \frac{1}{{\sqrt[3]{x}}}} \right)$$ with respect to $$x$$. Auxiliary Power Units. As per the power rule of integration, if we integrate x raised to the power n, then; xn dx = (xn+1/n+1) + C By this rule the above integration of squared term is justified, i.e.x2 dx. THE INTEGRATION OF EXPONENTIAL FUNCTIONS. 4 views, 0 likes, 0 loves, 0 comments, 0 shares, Facebook Watch Videos from : #_ Integration is a method of bringing different parts to form a whole. = xn ( (n+1) has got canceled). \[\int 4x^3 dx\] Use the power rule to differentiate each power function. & = - \frac{3}{4}.\frac{1}{x^4} + c \\ By the constant multiple rule of integration, = (-2/5) (x6/6) + C 05. \end{aligned} \], We integrate \( \int \frac{\sqrt[3]{x}}{4} dx\) as follows: \end{aligned} \], We integrate \(\int 2. Here is the Power Rule with some sample values. Let's look at a few more examples to get a better understanding of the power rule and its extended differentiation methods. Additionally, we will explore several examples with answers to understand the application of the power rule formula. For two functions, it may be stated in Lagrange's notation as. & = \frac{-1}{-1}x^{-1} +c \\ It is evaluated that the derivative of the expression x n + 1 + k is ( n + 1) x n. According to the inverse operation, the primitive or an anti-derivative of expression ( n + 1) x n is equal to x n + 1 + k. It can be written in mathematical form as follows. Integration by Parts Example 1. By doing this, we will have a single variable raised to a rational numerical exponent. Exponential functions occur frequently in physical sciences, so it can be very helpful to be able to integrate them. chain rule composite functions power functions power rule . \[\begin{aligned} \int 2 dx & = \int 2x^0 dx \\ For example, x5 dx = (x6) / 6 + C. Have questions on basic mathematical concepts? Proving the Power Rule by inverse operation. Find: \(\displaystyle\int \sqrt{x} + 4 \text{ dx}\). Product rule. As mentioned at the beginning of this section, exponential functions are used in many real-life applications. & = \frac{5}{2}\int x^{-3}dx \\ Power Rule Example Four www.statistica.com.au. Use the power rule formula detailed above to solve the exercises. power rule of integration is used to integrate the functions with exponents. :) Learn More In calculus, the product rule (or Leibniz rule [1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. & = \frac{1}{4} \times \frac{1}{\frac{1}{3}+1}.x^{\frac{1}{3}+1}+c \\ Scroll down the page for more examples and solutions. By the end of this section we'll know how to evaluate integrals like: Now, simplify the expression to find your final answer. \[\begin{aligned} \text{then} \quad F(x) &= \int a.x^n dx \\ We now look at integrals in which the integrand has more than one term. Now, lets look at how this kind of integral would be with skipping some of the more straightforward steps. We have some integration rules to find out the integral of . & = \frac{2}{3}.x^{\frac{3}{2}}+c \\ It is useful when finding the derivative of a function that is raised to the nth power. \[\begin{gathered} I = \frac{{{x^{ 2 + 1}}}}{{ 2 + 1}} 2x + c \\ \Rightarrow I = \frac{{{x^{ 1}}}}{{ 1}} 2x + c \\ \Rightarrow I = \frac{1}{x} 2x + c \\ \end{gathered} \]. Services; Thermo King Parts and Accessories; 24/7 . To illustrate, the formula is. Then, list down this form of power rule formula for our reference: Lets now convert the function from radical to exponential form: Then, lets determine the exponent of our variable. In this case, our exponent is $latex \frac{11}{29}$. & = \frac{6}{-4}x^{-4} + c \\ Important Notes on Power Rule of Integration: Example 1: What is the value of 2x3 + 1 dx? When a function is raised to some power then the rule used for integration is: fx.dx = (x n+1)/n+1 . arrow_forward Literature guides Concept explainers Writing guide Popular textbooks Popular high school textbooks Popular Q&A Business Accounting Economics Finance Leadership Management Marketing Operations Management Engineering Bioengineering Chemical Engineering Civil Engineering Computer Engineering Computer Science Electrical Engineering . Check out all of our online calculators here! \int 4 \sqrt[5]{x^4} dx & = \frac{20}{9}.\sqrt[5]{x^9} + c EXAMPLE 1 Find the derivative of f ( x) = x 2 e 2 x Solution EXAMPLE 2 What is the derivative of f ( x) = l n ( x) cos ( x)? If y = 2x + 7. or y = 2x - 8. or y = 2x + 100000. then for all cases dy/dx = 2 Hence we can say that anti-derivative or integral of 2 is 2x + C where C = any real constant. Power rule works for differentiating power functions. Take a look at the example to see how. Use the power rule formula detailed above to solve the exercises. & = \frac{3}{2}x^8 + c \end{aligned}\], We integrate \(-2x^5\) as follows: & = \frac{2}{-3+1}x^{-3+1}+c \\ & = \frac{1}{4}\times \frac{3}{4}.x^{\frac{4}{3}}+c \\ & = - \frac{3}{4x^4} + c \end{aligned}\], To integrate \(-\frac{1}{x^2}\) we use the fact that \(-\frac{1}{x^2} = -x^{-2}\): There is a different rule for dealing with functions like \(\dfrac{1}{x}\). \int 6.\sqrt{x} dx & = 4.\sqrt{x^3}+c Add a C at the end. It is not always necessary to compute derivatives directly from the definition. & = \frac{2}{\frac{1}{3}+1}.x^{\frac{1}{3}+1}+c \\ Let's first prove that this rule is the reverse of the power rule for differentiation. 3. The "Power Rule for Integration" Problem Pack has tips and tricks for working problems as well as plenty of practicewith full step-by-step solutions. & = - \frac{5}{4}.\frac{1}{x^2}+c \\ Thus, d/dx ((xn+1) / (n+1) + C) = xn and hence xn dx = (xn+1) / (n+1) + C. Hence, proved. We start by learning the power rule for integration formula, before watching a tutorial and working through some exercises. = (2x3)/3 - (3x2)/2 + C. We have a property of negative exponents that says 1/am = a-m. The power rule of integration is one of the integration rules that is used to integrate a term that has an exponent in it. & = x^{-1} + c \\ Get detailed solutions to your math problems with our Power Rule for Derivatives step-by-step calculator. This representation helps to convert a radical into exponent form. Nearly all of these integrals come down to two basic formulas: \int e^x . 15446 reads. \[\begin{aligned} \int \frac{2}{x^3} dx &= \int 2x^{-3} dx \\ \[\begin{aligned} \int \frac{\sqrt[3]{x}}{4} dx = 1/(n+1) [ (n + 1) xn+1-1] (by power rule of derivatives) Since this is a single variable raised to a numerical exponent, we can list down this form of power rule formula for our reference: Then, lets determine the exponent of our variable. & = \frac{5}{2} \times \frac{1}{-3+1}x^{-3+1}+c \\ Since this is a simple rational function, we can apply the laws of exponents to transform the rational form into its exponential form. Here it is formally: The Constant Multiple Rule for Integration tells you that it's okay to move a constant outside of an integral before you integrate. But this rule is used to find the integrals of non-zero constants and the integral of zero as well. Examples 7 Example: Evaluate Solution: Example: Evaluate Solution: 8. The general power rule is a special case of the chain rule. If you can write it with an exponents, you probably can apply the power rule. We also treat each of the "special cases" such as negatitive and fractional exponents to integrate functions involving roots and reciprocal powers of \(x\). Hence, $$\frac{d}{dx} (x^n) = \frac{d}{dx} (7x^{\frac{11}{29}})$$, $$ \frac{d}{dx} (x^n) = 7 \cdot \left[ \left(\frac{11}{29} \right) \cdot x^{\left(\frac{11}{29} \right)-1} \right]$$, $$\frac{d}{dx} (x^n) = 7 \cdot \left(\frac{11}{29} x^{-\frac{18}{29}} \right)$$, $$\frac{d}{dx} (x^n) = \frac{77}{29} x^{-\frac{18}{29}} $$, $$\frac{d}{dx} (x^n) = \frac{77}{29x^{\frac{18}{29}}}$$, $$f'(x) = \frac{77}{29 \hspace{2.3 pt} \sqrt[29]{x^{18}}}$$in radical form, Find the derivative of $latex f'(x) = \frac{1}{\sqrt{x^5}}$. mathematics statistica tutor spss perth statistic statistics parents services help. *Click on Open button to open and print to worksheet. As you will see, no matter how many fractions you are dealing with, the approach will stay the same. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Your email address will not be published. John Radford [BEng(Hons), MSc, DIC] d/dx ((xn+1) / (n+1) + C) = d/dx ((xn+1) / (n+1)) + d/dx (C) Find: \(\displaystyle\int \dfrac{1}{2}\sqrt[3]{x} + 5\sqrt[4]{x^3} \text{ dx}\). \(\displaystyle\int \sqrt{x} + 4 \text{ dx} = \displaystyle\int {x}^{\frac{1}{2}} + 4 \text{ dx}\). Important notes: This coursework is composed of Two-Parts that run-in succession. The rule may be extended or generalized to products of three or more functions, to a rule for higher-order . & = \frac{4}{\frac{4}{5}+1}.x^{\frac{4}{5}+1}+c \\ Solution Use substitution, setting u = x, and then du = 1dx. Find: \(\displaystyle\int \dfrac{3}{x^5} \dfrac{1}{4x^2} \text{ dx}\). & = \frac{1}{4} \int \sqrt[3]{x} dx \\ In this case, our exponent is $latex -\frac{5}{2}$. Now, by the power rule of integration, = 3 (x5/4) / (5/4) + C For instance: \(\int \begin{pmatrix} x^2 + x^3 \end{pmatrix}dx\). To apply the power rule of integration, the exponent of x can be any number (positive, 0, or negative) just other than -1. Find the derivative of f ( x) = 35 x 36 Choose an answer 1200 x 1225 x 35 1260 x 35 1296 x 36 Check What is the derivative of f ( x) = 40 x 30 + 20 x 10 5 x 5 2 + 2 ? Here are some examples of this rule: We know that integration is the reverse process of differentiation and if the integral of a function F(x) is f(x), then differentiating f(x) gives F(x) back. Before applying any calculus, you can rewrite the integral using the rule above. Ex) Derivative of 2 x 10 + 7 x 2 Derivative Of A Negative Power Example Ex) Derivative of 4 x 3 / 5 + 7 x 5 Find Derivative Rational Exponents Example Summary 1 - Integral of a power function: f(x) = x n . & = 6\times \frac{1}{\frac{1}{2}+1}.x^{\frac{1}{2}+1}+c \\ Then, list down this form of power rule formula for our reference: Lets now convert the function from rational to exponential form by applying the laws of exponents: $$\frac{d}{dx} (x^n) = \frac{d}{dx} (3x^{-15})$$, $$\frac{d}{dx} (x^n) = 3 \cdot (-15x^{(-15)-1})$$, Bringing the derived equation back into the rational form by applying the laws of exponents, we have, $$\frac{d}{dx} (x^n) = \frac{-45}{x^{16}}$$. For example, 1/x2 dx = x-2 dx and by integrating this using power rule, we get x-2 dx = (x-2+1)/(-2+1) + C = (x-1)/(-1) + C = -1/x + C. Here are some more examples: Note: We cannot integrate (1/x) dx using the power rule by writing it as x-1 dx. Because, if we apply the power rule for this, we get x0/0 + C. But x0/0 is not defined. Then, lets identify the transcendental function and the numerical exponent from the given problem: $$ \frac{d}{dx} (u^n) = \frac{d}{dx} (e^{-2x})$$, $$\frac{d}{dx} (u^n) = 2 \cdot (e^x)^{-2-1} \cdot e^x$$, Simplifying algebraically and applying the laws of exponents, we have, $$\frac{d}{dx} (u^n) = 2 \cdot (e^x)^{-3} \cdot e^x$$, $$\frac{d}{dx} (u^n) = \frac{2}{e^{2x}}$$. = (12 x5/4)/5 + C, The formula for power rule of integration says xn dx = (xn+1) / (n+1) + C, where. Hence. i.e., the power rule of integration rule can be applied for: Polynomial functions (like x 3, x 2, etc) Radical functions (like x, x, etc) as they can be written as exponents & = -x^{-2}+c \\ We can write x = x1/4. Lets work with one that is a little more messy with the fractions. Usually, the final answer can be written using exponents like we did here or with roots. The area of the region enclosed by the graph of functions is defined and calculated using integration.. Consider the function to be integrated. By doing this, we will have a single variable raised to a negative numerical exponent. & = \frac{3}{2}.x^{\frac{4}{3}}+c \\ & = \frac{6}{-5+1}x^{-5+1} + c \\ It is derived from the power rule of differentiation. So let's do a couple of examples just to make sure that that actually makes sense. \[\begin{gathered} I = \int {\left( {\frac{{{x^2}}}{{{x^4}}} 2\frac{{{x^4}}}{{{x^4}}}} \right)dx} \\ \Rightarrow I = \int {\left( {\frac{1}{{{x^2}}} 2} \right)dx} \\ \Rightarrow I = \int {{x^{ 2}}dx 2\int {dx} } \\ \end{gathered} \], Using the power rule of integration, we have If you have problems with these exercises, you can study the examples solved above. What is the derivative of $latex f(x) = 7 \sqrt[29]{x^{11}}$? Finally, dont forget to add the constant C. Finding the integral of a polynomial involves applying the power rule, along with some other properties of integrals. Power Rule of Integration In accordance with the power rule of integration, if y raised to the power n is integrated, the result is yn dy = (yn+1/n+1) + C Example: Integrate y 4 dy. The power rule of integration is used to integrate the functions with exponents. Basic Rules of Integration TechnologyUK. We start with the derivative of a power function, f ( x) = x n. Here n is a number of any kind: integer, rational, positive, negative, even irrational, as in x . Using the power rule of integration, we have. & = - \frac{5}{4}.x^{-2}+c\\ Then, sum/difference of derivatives will be applied to the whole polynomial function. = 6 (using power rule) (use the derivative calculator to solve). Part I runs from week 1 to week 6 and Part I & = \frac{2}{\frac{4}{3}}x^{\frac{4}{3}}+c \\ Thus, it is possible to integrate radicals using the power rule of integration. The basic power rule of integration is of the form. We have an \(x\) by itself and a constant. & = \int 6.x^{\frac{1}{2}}+c \\ Interested in learning more about the power rule? D. Power of a Function 8 The integral of a function raised to a power is found by the following steps: 1. Math - Calculus - DrOfEng Published May 10, 2022 6 Views. Integration, power rule, examples - Calculus. THE TRAPEZIUM RULE, Integration From A-level Maths Tutor www.a-levelmathstutor.com. Exploring the power rule of derivatives with examples. Note that this only works when the exponent is not 1. \[f(x) = \frac{a}{x^n}\] \int \sqrt{x} dx & = \frac{2}{3}.\sqrt{x^3}+c Basic examples of Integration rules. Choose an answer & = \frac{2}{0+1}x^{0+1} + c \\ \[\begin{aligned} \int \frac{5}{2x^3} dx & = \int \frac{5}{2}x^{-3} dx \\ radicals. The reverse power rule tells us how to integrate expressions of the form where : Basically, you increase the power by one and then divide by the power . Now that we've seen that we can integrate functions looking like \(f(x)=\frac{a}{x^n}\) using negative powers of \(x\), let's work through the exercise below. This one is a little different. (d/dx) ( 3 8) x 3 = ( 3 8) (d/dx) x 3. The general power rule states that this derivative is n times the function raised to the (n-1)th power times the derivative of the function. First and foremost, we need to identify the case and list the appropriate form of the power rule formula. \end{aligned}\], To integrate \(\frac{5}{2x^3}\), we use the fact that \(\frac{5}{2x^3} = \frac{5}{2}x^{-3}\): Since this is a radical function, we can apply the laws of exponents to transform the radical form into its exponential form. Exponential functions are those of the form f (x)=Ce^ {x} f (x) = C ex for a constant C C, and the linear shifts, inverses, and quotients of such functions. Add the constant of integration. Each algebraic term of the polynomial will use the basic power rule formula. & = - \frac{3}{2x^4} + c Practice your math skills and learn step by step with our math solver. Solution EXAMPLE 3 Find the derivative of f ( x) = x 3 sin 2 ( x) For example, f(x) = 2x2 - 3x is a polynomial function and we can apply the power rule and properties of integrals as shown below to integrate this. As you have seen, the power rule can be used to find simple integrals, but also much more complicated integrals. The first two examples contain exponential functions of different bases. The power rule of integration is one of the rules of integration and that is used to find the integral (in terms of a variable, say x) of powers of x. Using the formula detailed above, we can derive various variables, polynomials, or transcendental functions raised to a numerical exponent. exdx = ex + C axdx = ax lna + C Example 5.6.1: Finding an Antiderivative of an Exponential Function Find the antiderivative of the exponential function e x. & = \frac{1}{4} \times \frac{1}{\frac{4}{3}}x^{\frac{4}{3}}+c \\ Section 1-1 : Integration by Parts Let's start off with this section with a couple of integrals that we should already be able to do to get us started. \int \frac{5}{2x^3} dx & = - \frac{5}{4x^2}+c It is recommended for you to try to solve the sample problems yourself before looking at the solution so that you can practice and fully master this topic. Rules: Power Rule: Integration's power rule is the inverse of differentiation's power rule. The General Power Formula as shown in Chapter 1 is in the form $\displaystyle \int u^n \, du = \dfrac{u^{n+1}}{n+1} + C; \,\,\, n \neq -1$ Thus far integration has been confined to polynomial functions. Exponential functions can be integrated using the following formulas. \[\text{if} \quad f(x) = a.x^n\] \[\begin{gathered} I = \int {\left( {{x^{\frac{1}{3}}} + \frac{1}{{{x^{\frac{1}{3}}}}}} \right)dx} \\ \Rightarrow I = \int {\left[ {{x^{\frac{1}{3}}} + {x^{ \frac{1}{3}}}} \right]dx} \\ \Rightarrow I = \int {{x^{\frac{1}{3}}}dx + \int {{x^{ \frac{1}{3}}}dx} } \\ \Rightarrow I = \frac{{{x^{\frac{1}{3} + 1}}}}{{\frac{1}{3} + 1}} + \frac{{{x^{ \frac{1}{3} + 1}}}}{{ \frac{1}{3} + 1}} + c \\ \Rightarrow I = \frac{{{x^{\frac{4}{3}}}}}{{\frac{4}{3}}} + \frac{{{x^{\frac{2}{3}}}}}{{\frac{2}{3}}} + c \\ \Rightarrow I = \frac{3}{4}{x^{\frac{4}{3}}} + \frac{3}{2}{x^{\frac{2}{3}}} + c \\ \end{gathered} \], Your email address will not be published. \[\begin{aligned} \int - 4x^5 dx & = -\frac{4}{5+1}x^{5+1} +c \\ The power rule for integrals The power rule for integrals allows us to find the indefinite (and later the definite) integrals of a variety of functions like polynomials, functions involving roots, and even some rational functions. 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