Question

$\int \frac{1 + \cos 4x}{\cot x - \tan x} dx$

Answer 1

$-\frac{1}{8}\cos 4x + C$

Answer 2

The integral is simplified by applying trigonometric identities to both the numerator and the denominator.

1. Numerator $1 + \cos 4x$ is transformed to $2\cos^2(2x)$ using the half-angle identity.
2. Denominator $\cot x - \tan x$ is transformed to $2\cot 2x$ by converting to sine and cosine and then using double angle identities.
3. The integral then simplifies to $\int \frac{\cos^2(2x)}{\cot 2x} dx$, which further reduces to $\int \sin 2x \cos 2x dx$.
4. This product is converted to $\frac{1}{2}\sin 4x$ using the double angle identity for sine.
5. Finally, the integral of $\frac{1}{2}\sin 4x$ is evaluated directly.