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26.7: Appendix D. Some Important Definite Integrals

  • Page ID
    192036
  • We frequently need the values of the definite integrals below. These values are available in standard tables. Note that integrands involving even powers of the argument are even functions; integrands involving odd powers are odd functions. (A function, \(f(x)\), is even if \(f(x) = f(-x)\); it is odd if \(f(x) = -f(-x)\).) The integrals are given over the interval \(0 < x < \infty\). For integrands that are even functions, the integrals over the interval \(- \infty < x < \infty\) are twice the integrals over the interval \(0 < x < \infty\). For integrands that are odd functions, the integrals over the interval \(- \infty < x < \infty\) are zero.

    \[ \begin{array}{l} \int_0^{\infty} \text{exp} \left( -ax^2 \right) dx = \frac{1}{2} \sqrt{ \frac{\pi}{a}} \\ \int_0^{ \infty} x \text{ exp} \left( -ax^2 \right) dx = \frac{1}{2a} \\ \int_0^{ \infty} x^2 \text{ exp} \left( -ax^2 \right) dx = \frac{1}{4} \sqrt{\frac{\pi}{a^3}} \\ \int_0^{\infty} x^3 \text{ exp} \left( -ax^2 \right) dx = \frac{1}{2a^2} \\ \int_0^{ \infty} x^4 \text{ exp} \left( -ax^2 \right) dx = \frac{3}{8} \sqrt{ \frac{\pi}{a^5}} \\ \int_0^{ \infty} x^6 \text{ exp} \left( -ax^2 \right) dx = \frac{15}{16} \sqrt{ \frac{\pi}{a^7}} \end{array} \nonumber\]

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