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Chapter 5: Problem 4

Find a suitable substitution for evaluating \(\int \tan x \sec ^{2} x d x\) and explain your choice.

Short Answer

Expert verified
Answer: The substitution used to evaluate the integral is \(u = \tan x\). The final solution of the integral is \(\frac{1}{2}(\tan x)^2 + C\).

Step by step solution

01

Consider the given integral

The given integral is: $$\int \tan x \sec^2 x dx$$
02

Observe the integrand

Observe that the derivative of \(\tan x\) is \(\sec^2 x\). A potential suitable substitution would be to replace \(\tan x\) with a new variable, say \(u\). Thus, let us substitute: $$u = \tan x$$
03

Find the derivative of the substitution

Now, find the derivative of \(u\) with respect to x: $$\frac{d u}{d x} = \frac{d}{d x} \tan x = \sec^2 x$$ Now, multiply both sides by \(dx\) to find the integral form: $$d u = \sec^2 x d x$$
04

Substitute the new variable into the integral

Replace \(\tan x\) by \(u\) and \(\sec^2 x d x\) by \(du\) in the integral: $$\int u du$$
05

Integrate with respect to the new variable

The integral now becomes much simpler to evaluate. Further integrate \(u\) w.r.t. \(u\): $$\int u du = \frac{1}{2}u^2 + C$$
06

Replace the new variable with the original function

We started with the substitution \(u = \tan x\), so we replace \(u\) with \(\tan x\) to get the final answer: $$\frac{1}{2}u^2 + C = \frac{1}{2}(\tan x)^2 + C$$ The substitution chosen was to let \(u = \tan x\). The final solution for the integral is: $$\frac{1}{2}(\tan x)^2 + C$$

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