Evaluate the Integral integral of sin(x)^3 with respect to x

Problem

(∫_^)(sin3(x)*d(x))

Solution

1. Rewrite the integrand using the Pythagorean identity sin2(x)=1−cos2(x) to separate one factor of sin(x)

(∫_^)(sin2(x)*sin(x)*d(x))

(∫_^)((1−cos2(x))*sin(x)*d(x))

2. Apply a substitution by letting u=cos(x)

u=cos(x)

3. Calculate the differential d(u)

d(u)=−sin(x)*d(x)

−d(u)=sin(x)*d(x)

4. Substitute the variables u and d(u) into the integral.

(∫_^)((1−u2)*(−1)*d(u))

(∫_^)((u2−1)*d(u))

5. Integrate the polynomial with respect to u

(u3)/3−u+C

6. Back-substitute u=cos(x) to express the result in terms of x

cos3(x)/3−cos(x)+C

Final Answer

(∫_^)(sin3(x)*d(x))=1/3*cos3(x)−cos(x)+C

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