Evaluate the Integral

Problem

(∫_^)(sec(x)*(sec(x)+tan(x))*d(x))

Solution

1. Distribute the sec(x) term into the parentheses to rewrite the integrand.

(∫_^)((sec2(x)+sec(x)*tan(x))*d(x))

2. Split the integral into two separate integrals using the sum rule for integration.

(∫_^)(sec2(x)*d(x))+(∫_^)(sec(x)*tan(x)*d(x))

3. Apply the basic trigonometric integration rules, noting that the derivative of tan(x) is sec2(x) and the derivative of sec(x) is sec(x)*tan(x)

tan(x)+sec(x)+C

Final Answer

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

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