Evaluate the Integral

Problem

(∫_^)(cos2(x)*tan3(x)*d(x))

Solution

1. Rewrite the integrand by expressing tan(x) in terms of sin(x) and cos(x)

(∫_^)(cos2(x)sin3(x)/cos3(x)*d(x))

2. Simplify the expression by canceling the cos2(x) terms.

(∫_^)(sin3(x)/cos(x)*d(x))

3. Apply the Pythagorean identity sin2(x)=1−cos2(x) to rewrite the numerator.

(∫_^)(((1−cos2(x))*sin(x))/cos(x)*d(x))

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

(∫_^)((1−u2)/u*(−d(u)))

5. Distribute the negative sign and split the fraction into two terms.

(∫_^)((u−1/u)*d(u))

6. Integrate each term with respect to u

(u2)/2−ln(u)+C

7. Back-substitute u=cos(x) to find the final result in terms of x

cos2(x)/2−ln(cos(x))+C

Final Answer

(∫_^)(cos2(x)*tan3(x)*d(x))=cos2(x)/2−ln(cos(x))+C

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