/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 40

Evaluate the following integrals. $$\int \csc ^{10} x \cot x d x$$

Short Answer

Expert verified
Question: Evaluate the integral \(\int \csc ^{10} x \cot x d x\). Answer: \(-\frac{1}{10\sin^{10} x} + C\)

Step by step solution

01

Rewrite the integral using known trigonometric identities

The given integral is \(\int \csc ^{10} x \cot x d x\). We can rewrite this function using the trigonometric identities \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\): $$\int \csc^{10} x \cot x d x = \int \frac{\cos x}{\sin^{11} x} d x$$
02

Perform a substitution using an appropriate trigonometric function

To evaluate this integral, we can use a substitution. Let us define: $$u = \sin x$$ $$\frac{du}{dx} = \cos x$$ $$dx = \frac{du}{\cos x}$$ Now, we can rewrite the integral in terms of \(u\): $$\int \frac{\cos x}{\sin^{11} x} d x = \int \frac{1}{u^{11}} \frac{du}{\cos x}$$
03

Apply the integration by parts technique to evaluate the resulting integral

The integral now becomes: $$\int \frac{1}{u^{11}} du = -\frac{1}{10u^{10}} + C$$ Now, we substitute \(u = \sin x\) back into the expression to get the final answer: $$-\frac{1}{10\sin^{10} x} + C$$ So, the evaluated integral is: $$\int \csc ^{10} x \cot x d x = -\frac{1}{10\sin^{10} x} + C$$

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