Problem 62 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.