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  • Integrals of rational and irrational functions

Math formulas: Rational and irrational integrals

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0dxx2+a2=π2a\int^\infty_0 \frac{dx}{x^2+a^2} = \frac{\pi}{2a}
0xp1dx1+x=πsin(pπ), 0<p<1\int^\infty_0 \frac{x^{p-1}\,dx}{1+x} = \frac{\pi}{\sin (p\pi)} , ~ 0 < p < 1
0xmxn+an=πam+1nnsin[(m+1)π/n], 0<m+1<n \int^\infty_0 \frac{x^m}{x^n + a^n} = \frac{\pi a^{m + 1 -n}}{n\,\sin[(m+1)\pi/n]}, ~0 < m + 1 < n
0adxa2x2=π2 \int^a_0 \frac{dx}{\sqrt{a^2 - x^2}} = \frac{\pi}{2}
0aa2x2dx=πa24 \int^a_0 \sqrt{a^2 - x^2}\,dx = \frac{\pi\,a^2}{4}
0axm(anxn)pdx=am+1+np Γ[(m+1)/n] Γ(p+1)nΓ[(m+1)/n+p+1] \int^a_0 x^m \left(a^n - x^n\right)^p\,dx = \frac{a^{m+1+np}~\Gamma[(m+1)/n]~\Gamma(p+1) }{n\,\Gamma[(m+1)/n + p +1]}

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