Evaluate the Integral

Problem

(∫_^)(tan^3(x)*sec(x)*d(x))

Solution

1. Rewrite the integrand by splitting tan^3(x) into tan^2(x)*tan(x) to prepare for a substitution.

(∫_^)(tan^2(x)*sec(x)*tan(x)*d(x))

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

(∫_^)((sec^2(x)−1)*sec(x)*tan(x)*d(x))

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

(∫_^)((u^2−1)*d(u))

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

(u^3)/3−u+C

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

sec^3(x)/3−sec(x)+C

Final Answer

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

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