Evaluate the Integral integral of tan(x)^2 with respect to x

Problem

(∫_^)(tan2(x)*d(x))

Solution

1. Identify the trigonometric identity needed to rewrite the integrand, as there is no direct power rule for the tangent function.

2. Apply the identity tan2(x)=sec2(x)−1 to transform the integral into a form that is easier to evaluate.

(∫_^)((sec2(x)−1)*d(x))

3. Split the integral into two separate terms using the sum rule for integration.

(∫_^)(sec2(x)*d(x))−(∫_^)(1*d(x))

4. Integrate each term individually using standard integration rules, noting that the derivative of tan(x) is sec2(x)

tan(x)−x+C

Final Answer

(∫_^)(tan2(x)*d(x))=tan(x)−x+C

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