Evaluate the Integral

Problem

(∫_^)((earctan(x))/(1+x2)*d(x))

Solution

1. Identify a suitable substitution by looking for a function and its derivative within the integrand.

2. Let u=arctan(x)

3. Differentiate u with respect to x to find d(u) which gives d(u)/d(x)=1/(1+x2) or d(u)=1/(1+x2)*d(x)

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

(∫_^)(eu*d(u))

5. Integrate the exponential function with respect to u

eu+C

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

Final Answer

(∫_^)((earctan(x))/(1+x2)*d(x))=earctan(x)+C

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