Question

Find the integral. Use a computer algebra system to confirm your result. $$ \int \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} d x $$

Short answer

The integral of \( \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} dx \) is \( \ln|\sec x + \tan x| - 2\sin x + C \)

Step-by-step solution

Rewrite using Trigonometric Identity

The first step in solving this integral is to rewrite it using trigonometric identities. Because \( \sin^2x + \cos^2x = 1 \), we can express \( \sin^2x \) as \( 1 - \cos^2x \). So the integral becomes: \( \int \frac{(1 - 2\cos^2x)}{\cos x} dx \)

Separate the fraction

Next, separate the fraction into two separate fractions that can be integrated directly: \( \int \frac{1}{\cos x} dx - \int \frac{2\cos^2x}{\cos x} dx = \int \sec x dx - 2\int \cos x dx \)

Compute each integral

Now compute each of these integrals separately. The integral of secant x is ln|sec x + tan x| and the integral of cos x is sin x. This gives: \( \ln|\sec x + \tan x| - 2\sin x + C \)