/*! 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 34 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 34

Evaluate the following integrals. $$\int \sqrt{\tan x} \sec ^{4} x d x$$

Short Answer

Expert verified
Question: Evaluate the integral: \(\int \sqrt{\tan x}\sec ^{4}x dx\). Answer: \(\frac{2}{3}(\tan x)^{\frac{3}{2}} + C\)

Step by step solution

01

Apply Substitution Rule

Let's denote \(u = \tan x\). This implies that: $$\frac{d u}{d x} = \sec^2 x \qquad \Rightarrow \qquad d u = \sec^2 x d x$$ Now, rewrite the integral in terms of \(u\) using the substitution: $$\int \sqrt{\tan x} \sec ^{4} x d x = \int \sqrt{u} \sec ^{2} x \sec ^{2} x d x$$ By using the substitution, we get: $$\int \sqrt{u} \sec ^{2} x \sec ^{2} x d x = \int \sqrt{u} du$$
02

Evaluate New Integral

Now, we have the integral: $$\int \sqrt{u} du$$ Which can be expressed as: $$\int u^{\frac{1}{2}} du$$ Now, apply the power rule for integration: $$\int u^{\frac{1}{2}} du = \frac{u^{\frac{1}{2}+1}}{\frac{1}{2}+1} + C = \frac{2}{3}u^{\frac{3}{2}} + C$$
03

Substitute Back and Simplify

Finally, substitute back \(u = \tan x\) to get the answer in terms of \(x\): $$\frac{2}{3}u^{\frac{3}{2}} + C = \frac{2}{3}(\tan x)^{\frac{3}{2}} + C$$ So, the solution to the original integral is: $$\int \sqrt{\tan x}\sec ^{4}x dx = \frac{2}{3}(\tan x)^{\frac{3}{2}} + C$$

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