Evaluate the Integral integral of tan(x)^4 with respect to x

Problem

(∫_^)(tan^4(x)*d(x))

Solution

1. Use a trigonometric identity to rewrite the integrand by splitting tan^4(x) into tan^2(x)*tan^2(x) and substituting tan^2(x)=sec^2(x)−1

(∫_^)(tan^2(x)*(sec^2(x)−1)*d(x))

2. Distribute the tan^2(x) term across the parentheses.

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

3. Split the integral into two separate parts.

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

4. Apply u-substitution to the first integral by letting u=tan(x) which means d(u)=sec^2(x)*d(x)

(∫_^)(u^2*d(u))=(u^3)/3=tan^3(x)/3

5. Apply a trigonometric identity to the second integral, replacing tan^2(x) with sec^2(x)−1

(∫_^)((sec^2(x)−1)*d(x))

6. Integrate the second part using the standard integrals (∫_^)(sec^2(x)*d(x))=tan(x) and (∫_^)(1*d(x))=x

tan(x)−x

7. Combine the results from both parts and include the constant of integration C

tan^3(x)/3−(tan(x)−x)+C

8. Simplify the final expression by distributing the negative sign.

tan^3(x)/3−tan(x)+x+C

Final Answer

(∫_^)(tan^4(x)*d(x))=tan^3(x)/3−tan(x)+x+C

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