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

Problem

(∫_^)(cos(x)*d(x))

Solution

1. Rewrite the integrand by splitting cos(x) into cos(x)⋅cos(x)

(∫_^)(cos(x)*cos(x)*d(x))

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

(∫_^)((1−sin(x))2*cos(x)*d(x))

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

(∫_^)((1−u2)2*d(u))

4. Expand the binomial expression inside the integral.

(∫_^)((1−2*u2+u4)*d(u))

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

u−(2*u3)/3+(u5)/5+C

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

sin(x)−(2*sin(x))/3+sin(x)/5+C

Final Answer

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

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