Evaluate the Integral

Problem

(∫_^)(tan(x)*sec(x)*d(x))

Solution

1. Identify the structure of the integrand, noting that sec(x) is the derivative of tan(x)

2. Choose a substitution by letting u=tan(x)

3. Calculate the differential d(u) by differentiating u with respect to x which gives d(u)=sec(x)*d(x)

4. Substitute the expressions for u and d(u) into the integral to transform it into a power function.

(∫_^)(u3*d(u))

5. Apply the power rule for integration, which states (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

(u4)/4+C

6. Back-substitute the original expression tan(x) for u to obtain the final result.

Final Answer

(∫_^)(tan(x)*sec(x)*d(x))=tan(x)/4+C

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