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Calculus · Definite-Integrals

Integrate 1/(x^4 + 1) from -oo to oo

\int_{-\infty}^{\infty} \frac{1}{x^{4} + 1}\, dx

Step by step

  1. \int_{-\infty}^{\infty} \frac{1}{x^{4} + 1}\, dx

    First find an antiderivative F, then evaluate F(b) − F(a).

  2. \int \frac{1}{x^{4} + 1}\, dx = - \frac{\sqrt{2} \log{\left(x^{2} - \sqrt{2} x + 1 \right)}}{8} + \frac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x - 1 \right)}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4}

    This one needs a special technique; the computer algebra system gives the antiderivative directly.

  3. F(\infty) - F(-\infty) = \left(\text{NaN}\right) - \left(\text{NaN}\right)

    Fundamental theorem of calculus: plug in the limits.

  4. = \frac{\sqrt{2} \pi}{2} \approx 2.2214

    Simplify.

Reveal the answer
\frac{\sqrt{2} \pi}{2}