Evaluate the Integral

Problem

(∫_^)(tan3(x)*sec(x)*d(x))

Solution

1. Rewrite the integrand by splitting tan3(x) into tan2(x)*tan(x) to prepare for a substitution.

(∫_^)(tan2(x)*sec(x)*tan(x)*d(x))

2. Apply the trigonometric identity tan2(x)=sec2(x)−1 to express the integrand in terms of sec(x)

(∫_^)((sec2(x)−1)*sec(x)*tan(x)*d(x))

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

(∫_^)((u2−1)*d(u))

4. Integrate the polynomial with respect to u using the power rule.

(u3)/3−u+C

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

sec3(x)/3−sec(x)+C

Final Answer

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

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