Evaluate the Integral

Problem

(∫_^)((x2)/(1+x2)*d(x))

Solution

1. Rewrite the numerator by adding and subtracting 1 to create a term that matches the denominator.

(∫_^)((x2+1−1)/(1+x2)*d(x))

2. Split the fraction into two separate terms.

(∫_^)(((x2+1)/(1+x2)−1/(1+x2))*d(x))

3. Simplify the first term in the integrand.

(∫_^)((1−1/(1+x2))*d(x))

4. Apply the sum rule for integration to integrate each term individually.

(∫_^)(1*d(x))−(∫_^)(1/(1+x2)*d(x))

5. Evaluate the integrals using the power rule and the known derivative of the inverse tangent function.

x−arctan(x)+C

Final Answer

(∫_^)((x2)/(1+x2)*d(x))=x−arctan(x)+C

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