Find the integral of cos x · sin(5 sin x + 2) dx.

Step-by-step solution

1. 1

   Look for an inner function whose derivative is already present. The composite piece is $\sin\left(5\sin x+2\right)$, whose inner function is $5\sin x+2$. Its derivative is $5\cos x$, and a $\cos x$ is sitting outside the composition — exactly the pattern that substitution is designed for.

2. 2

   Make the substitution. Let

   $u=5\sin x+2\quad\Longrightarrow\quad du=5\cos x\,dx\quad\Longrightarrow\quad \cos x\,dx=\frac{1}{5}\,du$

   Solving for $\cos x\,dx$ (rather than for $dx$ alone) is what makes every $x$ disappear.

3. 3

   Rewrite the integral entirely in u.

   $\int \sin(u)\cdot\frac15\,du=\frac15\int\sin u\,du$

4. 4

   Integrate.

   $\frac15\int\sin u\,du=\frac15(-\cos u)+C=-\frac{\cos u}{5}+C$

5. 5

   Substitute back.

   $\boxed{-\frac{1}{5}\cos\left(5\sin x+2\right)+C}$

6. 6

   Check by differentiating. By the chain rule,

   $\frac{d}{dx}\left[-\frac15\cos\left(5\sin x+2\right)\right]=-\frac15\cdot\left(-\sin\left(5\sin x+2\right)\right)\cdot 5\cos x=\cos x\,\sin\left(5\sin x+2\right)$

   which is the original integrand ✓. Note the two factors of $5$ cancel — a good reminder that the $\tfrac15$ in the answer is not optional.