Evaluate the Integral

Problem

(∫_^)(sec3(x)*tan3(x)*d(x))

Solution

1. Rewrite the integrand to isolate a factor of sec(x)*tan(x) for substitution, since the power of tan(x) is odd.

(∫_^)(sec2(x)*tan2(x)*(sec(x)*tan(x))*d(x))

2. Use the identity tan2(x)=sec2(x)−1 to express the remaining tangent terms in terms of secant.

(∫_^)(sec2(x)*(sec2(x)−1)*sec(x)*tan(x)*d(x))

3. Distribute the sec2(x) term inside the parentheses.

(∫_^)((sec4(x)−sec2(x))*sec(x)*tan(x)*d(x))

4. Substitute u=sec(x) which implies d(u)=sec(x)*tan(x)*d(x)

(∫_^)((u4−u2)*d(u))

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

(u5)/5−(u3)/3+C

6. Back-substitute u=sec(x) to return to the original variable.

sec5(x)/5−sec3(x)/3+C

Final Answer

(∫_^)(sec3(x)*tan3(x)*d(x))=sec5(x)/5−sec3(x)/3+C

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