Evaluate the Integral

Problem

(∫_^)(sec2(x)*tan(x)*d(x))

Solution

1. Identify a suitable substitution by observing that the derivative of tan(x) is sec2(x)

2. Substitute u=tan(x) which implies that d(u)=sec2(x)*d(x)

3. Rewrite the integral in terms of u

(∫_^)(u*d(u))

4. Integrate using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

(u2)/2+C

5. Back-substitute u=tan(x) to express the result in terms of x

Final Answer

(∫_^)(sec2(x)*tan(x)*d(x))=tan2(x)/2+C

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