Integration Strategy — Question 8

Question 8

State the structural cue, then evaluate: $$\int\sin x\cos^4x\,dx.$$

Question 8 – Solution

Step 1: Identify the structural cue. The sine power is odd, and $\sin x\,dx$ is the differential of $-\cos x$. Keep the sine factor and use the cosine as the substitution.

Step 2: Substitute $u=\cos x$. Then $$du=-\sin x\,dx,\qquad \sin x\,dx=-du.$$ Step 3: Rewrite and integrate. $$\begin{align*} \int\sin x\cos^4x\,dx &=-\int u^4\,du\\ &=-\frac{u^5}{5}+C. \end{align*}$$ Step 4: Return to $x$. $$\boxed{-\frac15\cos^5x+C}$$