/*! 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 42 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 42

Evaluate the following integrals. $$\int \tan ^{5} \theta \sec ^{4} \theta d \theta$$

Short Answer

Expert verified
The final expression of the integral in terms of the original variable is $\frac{1}{9}(\sec^2(\theta) - 1)^3 \sec^3(\theta) + C$.

Step by step solution

01

Choose the substitution variable

Let \(u = \sec(\theta)\).
02

Differentiate the chosen variable

Calculate the derivative of \(u\) with respect to \(\theta\). $$\frac{du}{d\theta} = \frac{d}{d\theta}(\sec(\theta)) = \sec(\theta) \tan(\theta)$$
03

Rewrite dz in terms of du

\(\displaystyle dz = \sec(\theta) \tan(\theta) d \theta = u\tan(\theta) d\theta\) Now, solving for \(d\theta\), we get \(\displaystyle d\theta = \frac{dz}{u\tan(\theta)}\)
04

Rewrite the integral in terms of u

Replace \(\sec\) and \(\tan\) functions with \(u\), and \(d\theta\) with the expression we found in Step 3. $$\int \tan ^{5}\theta \sec ^{4}\theta d\theta = \int \tan ^{5}\theta u^{4} \frac{dz}{u\tan(\theta)}$$
05

Simplify the integral

Note that \(\tan^{2}(\theta) = u^2 - 1\), since \(\sec^{2}(\theta) = \tan^2(\theta) + 1\). So \(\tan^4(\theta) = (u^2 - 1)^2\). Then, $$\int \tan ^{5}\theta \sec ^{4}\theta d\theta = \int (u^2 - 1)^2 \tan(\theta) u^{3} \frac{dz}{u}$$
06

Integrate with respect to u

Eliminate the terms that cancel out in the integrand and integrate the remaining expression. $$ = \int (u^2 - 1)^2 u^{3} dz$$
07

Evaluate the integral and write the result in terms of the original variable

After evaluating the integral, substitute back the original variable. $$= \frac{1}{9}(u^2 - 1)^3 u^3 + C$$ Replace \(u\) by \(\sec(\theta)\): $$= \frac{1}{9}(\sec^2(\theta) - 1)^3 \sec^3(\theta) + C$$

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