Evaluate the definite integral of (x^2 + 1) / (x^4 + 1) dx from 0 to infinity.
Get the complete, step-by-step math solution for: "Evaluate the definite integral of (x^2 + 1) / (x^4 + 1) dx from 0 to infinity....". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Rewrite the integrand
To simplify the integrand, we divide both the numerator and the denominator by x^2$. This transformation is useful for integrals involving x^4 + 1 in the denominator, as it often leads to a form that can be integrated using substitution.
Step 2: Manipulate the denominator
We can rewrite the denominator x^2 + \frac{1}{x^2} using the algebraic identity (a-b)^2 = a^2 - 2ab + b^2$. Here, a=x and b=\frac{1}{x}$, so (x - \frac{1}{x})^2 = x^2 - 2(x)(\frac{1}{x}) + \frac{1}{x^2} = x^2 - 2 + \frac{1}{x^2}$. Therefore, x^2 + \frac{1}{x^2} = (x - \frac{1}{x})^2 + 2$.
Step 3: Apply substitution
We introduce a substitution u = x - \frac{1}{x}$. Differentiating u with respect to x gives du = (1 + \frac{1}{x^2}) dx$, which matches the numerator. We also need to change the limits of integration. As x approaches 0 from the positive side, u approaches -\infty$. As x approaches \infty$, u approaches \infty$.
Step 4: Integrate with respect to u
The integral transforms into a standard form \int \frac{1}{u^2 + a^2} du = \frac{1}{a} \tan^{-1}(\frac{u}{a}) + C$. Here, a^2 = 2$, so a = \sqrt{2}$. We then evaluate the definite integral using the new limits.
Step 5: Evaluate the definite integral
We evaluate the inverse tangent function at the limits. As u \to \infty$, \tan^{-1}(u) \to \frac{\pi}{2}$. As u \to -\infty$, \tan^{-1}(u) \to -\frac{\pi}{2}$. Substituting these values gives the final result.