Evaluate the definite integral of (x2+1) / (x4+1) dx from 0 to infinity.
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Step-by-Step Solution
Step 1: Rewrite the integrand
To simplify the integrand, we divide both the numerator and the denominator by x2. This transformation is useful for integrals involving x4+1 in the denominator, as it often leads to a form that can be integrated using substitution.
Step 2: Manipulate the denominator
We can rewrite the denominator x2+x21 using the algebraic identity (a-b)^2 = a2 - 2ab + b2 .Here,a=xandb=x1, so (x - x1)^2 = x2 - 2(x)(x1) + x21=x2 - 2 + x21 .Therefore,x2+x21=(x−x1)2+2.
Step 3: Apply substitution
We introduce a substitution u=x - x1 Differentiating u with respect to x gives du =(1+x21)dx, which matches the numerator. We also need to change the limits of integration. As x approaches 0 from the positive side, u approaches - ∞ .Asxapproaches∞, u approaches ∞.
Step 4: Integrate with respect to u
The integral transforms into a standard form ∫u2+a21du=a1tan−1(au)+C .Here,a2=2, so a=2. We then evaluate the definite integral using the new limits.
Step 5: Evaluate the definite integral
We evaluate the inverse tangent function at the limits. As u →∞ ,tan−1(u)→2π. As u →−∞ ,tan−1(u)→−2π. Substituting these values gives the final result.