Question

$intcos 4\theta d\theta$

Explanation:

Step1: Use substitution

Let $u = 4\theta$, then $du=4d\theta$, and $d\theta=\frac{1}{4}du$.

Step2: Rewrite the integral

The integral $\int\cos(4\theta)d\theta$ becomes $\frac{1}{4}\int\cos(u)du$.

Step3: Integrate $\cos(u)$

We know that $\int\cos(u)du=\sin(u)+C$. So, $\frac{1}{4}\int\cos(u)du=\frac{1}{4}\sin(u)+C$.

Step4: Substitute back $u = 4\theta$

We get $\frac{1}{4}\sin(4\theta)+C$.

Answer:

$\frac{1}{4}\sin(4\theta)+C$