Evaluate the Integral

Problem

(∫_^)(x/((x2+1)2)*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand. Let u=x2+1

2. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=2*x

3. Solve for the differential x*d(x) to substitute into the integral.

x*d(x)=1/2*d(u)

4. Substitute the expressions for u and x*d(x) into the original integral.

(∫_^)(1/(2*u2)*d(u))

5. Rewrite the integral using a negative exponent to prepare for the power rule.

1/2*(∫_^)(u(−2)*d(u))

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

1/2*((u(−1))/(−1))+C

7. Simplify the resulting expression.

−1/(2*u)+C

8. Back-substitute u=x2+1 to return to the original variable x

−1/(2*(x2+1))+C

Final Answer

(∫_^)(x/((x2+1)2)*d(x))=−1/(2*(x2+1))+C

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