This list contains some of the essential ready-to-use integrals. They are helpful in Quantum Mechanics. While solving probability or expectation values problems, these integrals come to great use. UPSC has asked many questions, which involves using such ready-to-use results. Physics is easy.
Important Integrals for Use in Quantum Mechanics
$ \int_{0}^{\infty} e^{-c x^2} dx= \frac{1}{2} \sqrt{\frac{\pi}{c}} $
$ \int_{0}^{\infty} x e^{-cx^2} dx = \frac{1}{2c} $
$ \int_{0}^{\infty} x^2 e^{-cx^2} dx = \frac{1}{4c} \sqrt{{\frac{\pi}{c}}} $
$ \int_{0}^{\infty} x^3 e^{-cx^2} dx = \frac{1}{2c^2} $
$ \int_{0}^{\infty} x^4 e^{-cx^2} dx = \frac{3}{8c^2} \sqrt{\frac{\pi}{c}} $
$c$ is any constant
Gamma function
$ \Gamma(n) = \int_{0}^{\infty} x^{n-1} e^{-x} dx $
$ \Gamma(\frac{-3}{2}) = \frac{4}{3} \sqrt{\pi} $
$ \Gamma(\frac{-1}{2}) = -2 \sqrt{\pi} $
$ \Gamma(\frac{1}{2}) = \sqrt{\pi} $
$ \Gamma(1) = 0! $
$ \Gamma(\frac{3}{2}) = \frac{1}{2} \sqrt{\pi} $
$ \Gamma(2) = 1! $
$ \Gamma(\frac{5}{2}) = \frac{3}{4} \sqrt{\pi} $
$ \Gamma(3) = 2! $
$ \Gamma(\frac{7}{2}) = \frac{15}{8} \sqrt{\pi} $
$ \Gamma(4) = 3! $
Gaussian Function
even sequence
$ \int_0^{\infty} e^{-ax^2} dx = (\frac{1}{2})(\frac{\pi}{a})^{\frac{1}{2}} $
$ \int_0^{\infty} x^2 e^{-ax^2} dx = (\frac{1}{4a})(\frac{\pi}{a})^{\frac{1}{2}} $
$ \int_0^{\infty} x^4 e^{-ax^2} dx = (\frac{3}{8a^2})(\frac{\pi}{a})^{\frac{1}{2}} $
$ \int_0^{\infty} x^{2n} e^{-ax^2} dx = \frac{1 \times 3 \times .... \times (2n-1)}{2^{n+1} \times {a^n}} (\frac{\pi}{a})^{\frac{1}{2}} $
odd sequence
$ \int_0^{\infty} x e^{-ax^2} dx = \frac{1}{2a} $
$ \int_0^{\infty} x^3 e^{-ax^2} dx = \frac{1}{2a^2} $
$ \int_0^{\infty} x^5 e^{-ax^2} dx = \frac{1}{2a^3} $
$ \int_0^{\infty} x^{2n+1} e^{-ax^2} dx = \frac{n!}{2}(\frac{1}{a^{n+1}}) $
$ \int_0^{\infty} e^{ax} dx = \frac{1}{a} $, where $a<0$
$ \int_0^{\infty} e^{-ax^2} dx = \frac{1}{a} \sqrt{\frac{\pi}{a}} $, where $a>0$
$ \int_{-\infty}^{\infty} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} $, where $a>0$
$ \int_{-\infty}^{\infty} e^{-ax^2} e^{-2bx} dx = \sqrt{\frac{\pi}{a}} e^{\frac{b^2}{a}} $, where $a>0$
$ \int_{-\infty}^{\infty} x e^{-a(x-b)} dx = b \sqrt{\frac{\pi}{a}} $
$ \int_{-\infty}^{\infty} x^2 e^{-ax^2} dx = (\frac{1}{2}) \sqrt{\frac{\pi}{a^3}} $
Primitive of $ e^{ax} \sin bx $
$ \int e^{ax} \space \sin bx \space dx = \frac{e^{ax}(a \space \sin bx - b \space \cos bx)}{a^2 + b^2} + C $
Primitive of $ e^{ax} \cos bx $
$ \int e^{ax} \space \cos bx \space dx = \frac{e^{ax}(a \space \cos bx + b \space \sin bx)}{a^2 + b^2} + C $
odd function: $ f(-x) = -f(x)$
even function: $ f(-x) = f(x)$
function multiplications
odd * odd = even
even * even = even
odd * even = odd
The integral of any even function between the limits $ -\infty $ to $ \infty $ is twice the integral from $0$ to $ \infty $.
The integral of any odd function between the limits $ -\infty$ to $ \infty $ is zero.