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Chapter 4: Problem 62

State the integration formula you would use to perform the integration. Do not integrate. $$ \int \frac{\sec ^{2} x}{\tan x} d x $$

Short Answer

Expert verified
The integral formula applicable to the function is \(\int f'(x)/f(x) dx = ln|f(x)| + C\). Here, \(f(x) = \tan(x)\) and its derivative \(f'(x) = \sec^2(x)\). So the answer is, use the logarithm of a function rule for integration.

Step by step solution

01

Identify the numerator

The numerator of the function to be integrated is \(\sec^2(x)\). The integral of \(\sec^2(x)\) is \(\tan(x)\), which is a known result.
02

Identify the denominator

The denominator of the function to be integrated is \(\tan(x)\). Usually when you find the derivative or integral of a function that is a fraction, it is helpful when the derivative of the numerator is equivalent to the denominator or vice versa.
03

Match with Integral Formula

We can match our function with the integral formula \(\int \frac{1}{g'(x)} dx = ln |g(x)| + C\), where \(g'(x)\) is the derivative of some function \(g(x)\). In our case, \(g(x) = \sec^2(x)\) and its derivative \(g'(x)\) is \(2\sec(x)\tan(x)\). This doesn't quite match our given function to be integrated.
04

Correct Matching Integral Formula

On second examination, this matches the integral formula \(\int f'(x) / f(x) dx = ln|f(x)| + C\). Here, \(f(x) = \tan(x)\) and its derivative \(f'(x) = \sec^2(x)\), which matches the components of our function to be integrated exactly.

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