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

Problem

(∫_^)(cos(x)*d(x))

Solution

1. Rewrite the integrand using a trigonometric identity to separate one cos(x) factor.

(∫_^)(cos(x)*cos(x)*d(x))

2. Substitute the identity cos(x)=1−sin(x) into the integral.

(∫_^)((1−sin(x))*cos(x)*d(x))

3. Apply u-substitution by letting u=sin(x) which implies d(u)=cos(x)*d(x)

(∫_^)((1−u2)*d(u))

4. Integrate the polynomial with respect to u

u−(u3)/3+C

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

sin(x)−sin(x)/3+C

Final Answer

(∫_^)(cos(x)*d(x))=sin(x)−sin(x)/3+C

Want more problems? Check here!