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Evaluate the following indefinite integral \int (x+9)^{-4} dx

Question:

Evaluate the following indefinite integral {eq}\int (x+9)^{-4} dx {/eq}

Indefinite Integral in Calculus:

Indefinite integration is used to find out the antiderivative of a function.

The integrand is given with its derivative, so we'll solve this problem by using a process of substitution and it will turn the integrand into a function that is easier to integrate.

To integrate power functions we use the integration power rule {eq}\displaystyle \int x^n dx= \dfrac{x^{n+1}}{n+1}+C. {/eq}

Answer and Explanation:

We are given:

{eq}\displaystyle \int (x+9)^{-4} dx {/eq}


Apply u-substitution {eq}u = x+9 \rightarrow \ du = \ dx {/eq}

{eq}= \displaystyle \int u^{-4} \ du {/eq}


Apply integral power rule:

{eq}= \displaystyle \dfrac{u^{-4+1}}{-4+1}+C {/eq}

{eq}= \displaystyle \dfrac{u^{-3}}{-3}+C {/eq}

{eq}= \displaystyle - \dfrac{1}{3u^3}+C {/eq}


Substitute back {eq}u = x+9 {/eq}

{eq}= \displaystyle - \dfrac{1}{3 (x+9)^3}+C {/eq}


Therefore the solution is:

{eq}{\boxed{ \displaystyle \int (x+9)^{-4} dx =- \dfrac{1}{3 (x+9)^3}+C}} {/eq}


Learn more about this topic:

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Indefinite Integrals as Anti Derivatives

from Math 104: Calculus

Chapter 12 / Lesson 11
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