Evaluate the Integral

Problem

(∫_^)(x/(x^4+2*x^2+1)*d(x))

Solution

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

(∫_^)(x/((x^2+1)^2)*d(x))

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

u=x^2+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*u^2)*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 (∫_^)(u^n*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*(x^2+1))+C

Final Answer

(∫_^)(x/(x^4+2*x^2+1)*d(x))=−1/(2*(x^2+1))+C

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