Find the Integral sin(x)^3

Problem

(∫_^)(sin3(x)*d(x))

Solution

1. Use a trigonometric identity to rewrite the integrand by splitting sin3(x) into sin2(x)*sin(x)

(∫_^)(sin2(x)*sin(x)*d(x))

2. Apply the Pythagorean identity sin2(x)=1−cos2(x) to express the integrand in terms of cos(x)

(∫_^)((1−cos2(x))*sin(x)*d(x))

3. Use u-substitution by letting u=cos(x)

u=cos(x)

4. Calculate the differential d(u) to substitute for d(x)

d(u)=−sin(x)*d(x)

−d(u)=sin(x)*d(x)

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

(∫_^)(−(1−u2)*d(u))

6. Distribute the negative sign to simplify the integrand.

(∫_^)((u2−1)*d(u))

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

(u3)/3−u+C

8. Substitute back u=cos(x) to get the final result in terms of x

cos3(x)/3−cos(x)+C

Final Answer

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

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