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

Problem

(∫_^)(x*sec2(x)*d(x))

Solution

1. Identify the method of integration by parts, which uses the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Assign the variables for integration by parts by letting u=x and d(v)=sec2(x)*d(x)

3. Differentiate u to find d(u)=d(x)

4. Integrate d(v) to find v=tan(x)

5. Substitute these values into the integration by parts formula:

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

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

7. Combine the terms and add the constant of integration C

Final Answer

(∫_^)(x*sec2(x)*d(x))=x*tan(x)−ln(sec(x))+C

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