Evaluate the Integral integral of tan(x)^5 with respect to x

Problem

(∫_^)(tan5(x)*d(x))

Solution

1. Use the trigonometric identity tan2(x)=sec2(x)−1 to rewrite the integral.

(∫_^)(tan3(x)*(sec2(x)−1)*d(x))

2. Distribute the tan3(x) term to split the integral into two parts.

(∫_^)(tan3(x)*sec2(x)*d(x))−(∫_^)(tan3(x)*d(x))

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

(∫_^)(u3*d(u))=tan4(x)/4

4. Rewrite the second integral (∫_^)(tan3(x)*d(x)) using the same identity tan2(x)=sec2(x)−1

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

5. Evaluate the remaining integrals using u=tan(x) for the first term and the standard integral formula (∫_^)(tan(x)*d(x))=ln(sec(x))

(∫_^)(tan(x)*sec2(x)*d(x))=tan2(x)/2

(∫_^)(tan(x)*d(x))=ln(sec(x))

6. Combine all results and distribute the negative sign from step 2.

tan4(x)/4−(tan2(x)/2−ln(sec(x)))+C

7. Simplify the final expression.

tan4(x)/4−tan2(x)/2+ln(sec(x))+C

Final Answer

(∫_^)(tan5(x)*d(x))=tan4(x)/4−tan2(x)/2+ln(sec(x))+C

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