Evaluate the Integral integral of tan(x)^3 with respect to x

Problem

(∫_^)(tan3(x)*d(x))

Solution

1. Use a trigonometric identity to rewrite the integrand by splitting tan3(x) into tan(x)*tan2(x)

(∫_^)(tan(x)*tan2(x)*d(x))

2. Substitute the identity tan2(x)=sec2(x)−1 into the integral.

(∫_^)(tan(x)*(sec2(x)−1)*d(x))

3. Distribute the tan(x) term to create two separate integrals.

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

4. Apply u-substitution to the first integral by letting u=tan(x) which means d(u)=sec2(x)*d(x)

(∫_^)(u*d(u))−(∫_^)(tan(x)*d(x))

5. Integrate both terms using the power rule for u and the standard integral formula for tan(x) which is ln(sec(x))

(u2)/2−ln(sec(x))+C

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

tan2(x)/2−ln(sec(x))+C

Final Answer

(∫_^)(tan3(x)*d(x))=tan2(x)/2−ln(sec(x))+C

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