Definite integral formula An integral with upper and lower limits is a Definite Integral. A Riemann integral is a definite integral where x is restricted to lie on the real line.

$${\int }_a^{\mathrm{\infty }}f(x)dx=\underset{b\to \mathrm{\infty }}{lim}[{\int }_a^{b}f(x)dx]$$

$${\int }_a^{b}f(x)dx=F(b)-F(a)$$

a and ∞, b are the lower and upper limits, F(a) is the lower limit value of the integral, F(b) is the upper limit value of the integral.

Definite Integrals Rational or Irrational Expression

• $\begin{array}{l}{\int }_a^{\mathrm{\infty }}\frac{dx}{x^2+a^2}=\frac{\pi }{2a}\end{array}$
• $\begin{array}{l}{\int }_a^{\mathrm{\infty }}\frac{x^{m}dx}{x^n+a^n}=\frac{\pi a^{m-n+1}}{n\sin \left(\frac{(m+1)\pi }{n}\right)},0<m+1<n\end{array}$
• $\begin{array}{l}{\int }_a^{\mathrm{\infty }}\frac{x^{p-1}dx}{1+x}=\frac{\pi }{\sin (p\pi )},0<p<1\end{array}$
• $\begin{array}{l}{\int }_a^{\mathrm{\infty }}\frac{x^{m}dx}{1+2x\cos \beta +x^2}=\frac{\pi \sin (m\beta )}{\sin (m\pi )\sin \beta }\end{array}$
• $\begin{array}{l}{\int }_a^{\mathrm{\infty }}\frac{dx}{\sqrt{a^2-x^2}}=\frac{\pi }{2}\end{array}$
• $\begin{array}{l}{\int }_a^{\mathrm{\infty }}\sqrt{a^2-x^2}dx=\frac{\pi a^2}{4}\end{array}$

Definite integrals of Trigonometric Functions

• $\begin{array}{l}{\int }_0^{\pi }\sin (mx)\sin (nx)dx=\{\begin{array}{cc}0& ifm\ne n\\ \frac{\pi }{2}& ifm=n\end{array}m,npositiveintegers\end{array}$
• $\begin{array}{l}{\int }_0^{\pi }\cos (mx)\cos (nx)dx=\{\begin{array}{cc}0& ifm\ne n\\ \frac{\pi }{2}& ifm=n\end{array}m,npositiveintegers\end{array}$
• $\begin{array}{l}{\int }_0^{\pi }\sin (mx)\cos (nx)dx=\{\begin{array}{cc}0& ifm+neven\\ \frac{2m}{m^2-n^2}& ifm+nodd\end{array}m,nintegers\end{array}$