Evaluate the Integral integral of sin(x)^5 with respect to x

Problem

(∫_^)(sin(x)*d(x))

Solution

1. Rewrite the integrand by splitting one factor of sin(x) to prepare for a substitution.

(∫_^)(sin(x)*sin(x)*d(x))

2. Apply the Pythagorean identity sin(x)=1−cos(x) to express the even power of sine in terms of cosine.

(∫_^)((1−cos(x))2*sin(x)*d(x))

3. Substitute u=cos(x) which implies d(u)=−sin(x)*d(x) or −d(u)=sin(x)*d(x)

(∫_^)((1−u2)2*(−1)*d(u))

4. Expand the squared binomial inside the integral.

−(∫_^)((1−2*u2+u4)*d(u))

5. Integrate term by term with respect to u

−(u−(2*u3)/3+(u5)/5)+C

6. Distribute the negative sign and substitute cos(x) back in for u

−cos(x)+(2*cos(x))/3−cos(x)/5+C

Final Answer

(∫_^)(sin(x)*d(x))=−cos(x)+(2*cos(x))/3−cos(x)/5+C

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