# 16.10: Definite integrals

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• $$\int\limits_{0}^{\infty }xe^{-x^2}dx=\frac{1}{2}$$
• $$\int\limits_{0}^{\infty }e^{-ax}dx=\frac{1}{a}$$,$$a>0$$
• $$\int\limits_{0}^{\infty }\sqrt{x}e^{-ax}dx=\frac{1}{2a}\sqrt{\frac{\pi}{a}}$$
• $$\int\limits_{0}^{\infty }x^{2n+1}e^{-ax^2}dx=\frac{n!}{2a^{n+1}}$$,$$a>0$$
• $$\int\limits_{0}^{\infty }x^{2n}e^{-ax^2}dx=\frac{1.3.5...(2n-1)}{2^{n+1}a^{n}}\sqrt{\frac{\pi}{a}}$$
• $$\int\limits_{0}^{\infty }x^{n}e^{-ax}dx=\frac{n!}{a^{n+1}}$$, $$a>0$$, $$n$$ positive integer

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