We have to evaluate the integral: ∫ sin1x8dxx9 This integral will be solved by substitution method…

Answered: ∫sin(1/x8)dx/x9

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter1: Equations And Inequalities

Section1.CT: Chapter Test

Problem 5CT: Solve for x: z=x.

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∫sin(1/x⁸)dx/x⁹

Expert Solution

Step 1

We have to evaluate the integral:

$\int \frac{\sin (\frac{1}{x^{8}}) d x}{x^{9}}$

This integral will be solved by substitution method since derivative of $\frac{1}{x^{8}}$ is present in the integral.

Let $t = \frac{1}{x^{8}}$

differentiating both sides with respect to $' x'$ , we get

$\begin{array}{rcl} \frac{d t}{d x} & = & \frac{d (\frac{1}{x^{8}})}{d x} \\ = & \frac{d x^{- 8}}{d x} \\ = & - 8 x^{- 8 - 1} \\ = & - 8 x^{- 9} \\ \frac{d t}{d x} & = & \frac{- 8}{x^{9}} \\ \frac{- d t}{8} & = & \frac{d x}{x^{9}} \end{array}$

We have used the derivative formula, $\frac{d x^{n}}{d x} = n x^{n - 1}$

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