/*! 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 37 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 37

Evaluate the following integrals. $$\int \sec ^{2} x \tan ^{1 / 2} x d x$$

Short Answer

Expert verified
Question: Evaluate the integral \(\int \sec ^{2}(x) \tan ^{1 / 2}(x) d x\). Answer: \(\frac{(\tan x)^{3/2}}{(3/2)} + C\)

Step by step solution

01

Perform substitution

Let u = \(\tan (x)\).
02

Find du in terms of dx

Differentiate u with respect to x to find du/dx: $$ \frac{d u}{d x} = \frac{d}{d x}\tan (x) \Rightarrow \mathrm{d} u = \sec ^{2}(x) \mathrm{d} x $$
03

Rewrite the integral in terms of u

Substitute u and du into the given integral: $$ \int \sec ^{2}(x) \tan ^{1 / 2}(x) d x = \int u^{1/2} d u $$
04

Evaluate the integral

Find the antiderivative of u^1/2 with respect to u: $$ \int u^{1/2} d u = \frac{u^{3/2}}{(3/2)} + C $$
05

Substitute back x value

Replace u with \(\tan(x)\) to get the final solution: $$ \frac{(\tan x)^{3/2}}{(3/2)} + C $$ The evaluated integral is: \(\frac{(\tan x)^{3/2}}{(3/2)} + C\).

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