In mathematics, the definite integral is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.
The fundamental theorem of calculus establishes the relationship between indefinite and definite integrals and introduces a technique for evaluating definite integrals.
If the interval is infinite the definite integral is called an improper integral and defined by using appropriate limiting procedures.
for example: A constant, such pi, that may be defined by the integral of an algebraic function over an algebraic domain is known as a period.
The following is a list of some of the most common or interesting definite integrals.
For a list of indefinite integrals see List of indefinite integrals.
sin a x
sinh b x
π
tanh
a π
{\displaystyle \int _{0}^{\infty }{\frac {\sin ax}{\sinh bx}}\ dx={\frac {\pi }{2b}}\tanh {\frac {a\pi }{2b}}}
cos a x
cosh b x
π
a π
{\displaystyle \int _{0}^{\infty }{\frac {\cos ax}{\cosh bx}}\ dx={\frac {\pi }{2b}}\cdot {\frac {1}{\cosh {\frac {a\pi }{2b}}}}}
sinh a x
π
{\displaystyle \int _{0}^{\infty }{\frac {x}{\sinh ax}}\ dx={\frac {\pi ^{2}}{4a^{2}}}}
sinh a x
π a
{\displaystyle \int _{0}^{\infty }{\frac {x^{2n+1}}{\sinh ax}}\ dx=c_{2n+1}\left({\frac {\pi }{a}}\right)^{2(n+1)},\quad c_{2n+1}={\frac {(-1)^{n}}{2}}\left({\frac {1}{2}}-\sum _{k=0}^{n-1}(-1)^{k}{2n+1 \choose 2k+1}c_{2k+1}\right),\quad c_{1}={\frac {1}{4}}}
π
{\displaystyle \int _{0}^{\infty }{\frac {1}{\cosh ax}}\ dx={\frac {\pi }{2a}}}
π a
{\displaystyle \int _{0}^{\infty }{\frac {x^{2n}}{\cosh ax}}\ dx=d_{2n}\left({\frac {\pi }{a}}\right)^{2n+1},\quad d_{2n}={\frac {(-1)^{n}}{2}}\left({\frac {1}{4^{n}}}-\sum _{k=0}^{n-1}(-1)^{k}{2n \choose 2k}d_{2k}\right),\quad d_{0}={\frac {1}{2}}}
{\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\ dx=\left(\lim _{x\to 0}f(x)-\lim _{x\to \infty }f(x)\right)\ln \left({\frac {b}{a}}\right)}
holds if the integral exists and
is continuous.