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

Problem

(∫_^)(x*tan2(x)*d(x))

Solution

1. Apply a trigonometric identity to rewrite the integrand using the identity tan2(x)=sec2(x)−1

(∫_^)(x*(sec2(x)−1)*d(x))

2. Distribute the variable x across the terms inside the integral.

(∫_^)((x*sec2(x)−x)*d(x))

3. Split the integral into two separate parts using the linearity property of integration.

(∫_^)(x*sec2(x)*d(x))−(∫_^)(x*d(x))

4. Apply integration by parts to the first integral (∫_^)(x*sec2(x)*d(x)) by setting u=x and d(v)=sec2(x)*d(x)

d(u)=d(x)

v=tan(x)

5. Substitute into the integration by parts formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

x*tan(x)−(∫_^)(tan(x)*d(x))

6. Evaluate the integral of tan(x) which is ln(sec(x))

x*tan(x)−ln(sec(x))

7. Evaluate the second integral (∫_^)(x*d(x)) using the power rule.

(∫_^)(x*d(x))=(x2)/2

8. Combine the results and add the constant of integration C

x*tan(x)−ln(sec(x))−(x2)/2+C

Final Answer

(∫_^)(x*tan2(x)*d(x))=x*tan(x)−ln(sec(x))−(x2)/2+C

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