Evaluate the Integral

Problem

(∫_0^π/2)(sin5(x)*d(x))

Solution

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

(∫_0^π/2)(sin4(x)*sin(x)*d(x))

2. Apply the Pythagorean identity sin2(x)=1−cos2(x) to express the even power of sine in terms of cosine.

(∫_0^π/2)((1−cos2(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)

u=cos(x)

4. Change the limits of integration based on the substitution: when x=0 u=cos(0)=1 when x=π/2 u=cos(π/2)=0

(∫_1^0)((1−u2)2*(−d(u)))

5. Simplify the integral by using the negative sign to swap the limits of integration.

(∫_0^1)((1−u2)2*d(u))

6. Expand the binomial expression inside the integral.

(∫_0^1)((1−2*u2+u4)*d(u))

7. Integrate each term with respect to u using the power rule.

[u−(2*u3)/3+(u5)/5]10

8. Evaluate the expression at the upper and lower limits.

(1−2/3+1/5)−(0)

9. Calculate the final numerical value by finding a common denominator.

15/15−10/15+3/15=8/15

Final Answer

(∫_0^π/2)(sin5(x)*d(x))=8/15

Want more problems? Check here!