Evaluate the integral of (x2+1) / (x4+1) dx from 0 to infinity.
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Step-by-Step Solution
Step 1: Simplify the integrand
To make the numerator and denominator suitable for substitution, we divide both the numerator and the denominator by x2. This is a standard technique for integrals involving x4+1 or x4+ax2+1 type denominators.
Step 2: Rewrite the denominator
We rewrite the denominator in terms of x−x1 by using the algebraic identity (a−b)2=a2−2ab+b2. In this case, a=x and b=x1, so a2+b2=(a−b)2+2ab becomes x2+x21=(x−x1)2+2⋅x⋅x1=(x−x1)2+2.
Step 3: Perform substitution
Let us substitute u=x−x1. Then, we find the differential du by differentiating u with respect to x. The derivative of x is 1, and the derivative of −x1 (or −x−1) is −(−1)x−2=x21. This means du=(1+x21)dx exactly matches our modified numerator.
Step 4: Change limits of integration
Since we are changing the variable of integration from x to u, we must also change the limits of integration. As x approaches 0 from the positive side, u approaches −∞. As x approaches ∞, u also approaches ∞.
Step 5: Substitute into the integral
Substitute u and du into the integral, along with the new limits. The numerator becomes du, and the denominator becomes u2+2. The integral is now a standard form integral.
Step 6: Evaluate the integral
This integral is of the form ∫a2+x21dx=a1arctan(ax). Here, x is u and a2 is 2, so a=2. We apply this formula to evaluate the definite integral with the new limits.
Step 7: Apply the limits of integration
We evaluate the arctan function at the upper and lower limits. As the argument of arctan approaches ∞, the function approaches 2π. As the argument approaches −∞, it approaches −2π. Subtracting the lower limit from the upper limit, we find the value of the integral.