Evaluate the Integral

Problem

(∫_^)((etan(x))/cos2(x)*d(x))

Solution

1. Rewrite the integrand using the trigonometric identity 1/cos2(x)=sec2(x)

(∫_^)(etan(x)*sec2(x)*d(x))

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

u=tan(x)

3. Differentiate u with respect to x to find d(u)

d(u)/d(x)=sec2(x)

d(u)=sec2(x)*d(x)

4. Substitute u and d(u) into the integral.

(∫_^)(eu*d(u))

5. Integrate the exponential function with respect to u

eu+C

6. Back-substitute the original expression for u

etan(x)+C

Final Answer

(∫_^)((etan(x))/cos2(x)*d(x))=etan(x)+C

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