Evaluate the Integral

Problem

(∫_^)(sec2(x)/tan(x)*d(x))

Solution

1. Identify a suitable substitution by noticing that the derivative of the denominator, tan(x) is exactly the numerator, sec2(x)

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

3. Rewrite the integral in terms of u

(∫_^)(1/u*d(u))

4. Integrate using the natural logarithm rule:

ln(u)+C

5. Back-substitute the original expression for u to find the final result in terms of x

Final Answer

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

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