Evaluate the integral of sin(x)/sin^2(4 cos(x)) dx.

Step-by-step solution

1. 1

   Spot the inner function and its derivative. The awkward part is the composite $\sin^{2}(4\cos x)$, whose inner function is $4\cos x$. Its derivative is $-4\sin x$, and a factor $\sin x$ is already sitting in the numerator, so the substitution $u=4\cos x$ will clear the integral completely.

2. 2

   Carry out the substitution. From $u=4\cos x$ we get $du=-4\sin x\,dx$, hence $\sin x\,dx=-\frac14\,du$, and $\int\frac{\sin x}{\sin^{2}(4\cos x)}\,dx=-\frac14\int\frac{du}{\sin^{2}u}=-\frac14\int\csc^{2}u\,du .$

3. 3

   Use the standard cosecant-squared antiderivative. Because $\frac{d}{du}\cot u=-\csc^{2}u$, we have $\int\csc^{2}u\,du=-\cot u+C$, so $-\frac14\int\csc^{2}u\,du=\frac14\cot u+C .$

4. 4

   Return to the original variable. Substituting $u=4\cos x$ back gives $\int\frac{\sin x}{\sin^{2}(4\cos x)}\,dx=\frac14\cot(4\cos x)+C .$

5. 5

   Verify by differentiating. $\frac{d}{dx}\left[\frac14\cot(4\cos x)\right]=\frac14\left(-\csc^{2}(4\cos x)\right)\cdot(-4\sin x)=\frac{\sin x}{\sin^{2}(4\cos x)}$, which is the original integrand, so the antiderivative is correct on every interval where $\sin(4\cos x)\ne 0$.