Evaluate the Integral

Problem

(∫_^)(x/(x4+2*x2+1)*d(x))

Solution

1. Recognize the denominator as a perfect square trinomial of the form (a2+2*a*b+b2)=(a+b)2

(∫_^)(x/((x2+1)2)*d(x))

2. Choose a substitution to simplify the expression. Let u=x2+1

u=x2+1

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

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

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

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

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

(∫_^)(1/(2*u2)*d(u))

5. Rewrite the integrand using a negative exponent to prepare for integration.

1/2*(∫_^)(u(−2)*d(u))

6. Apply the power rule for integration, which states (∫_^)(un*d(u))=(u(n+1))/(n+1)

1/2*((u(−1))/(−1))+C

−1/(2*u)+C

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

−1/(2*(x2+1))+C

Final Answer

(∫_^)(x/(x4+2*x2+1)*d(x))=−1/(2*(x2+1))+C

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