In mathematics, the definite integral
∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx}
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 ∞ f ( x ) d x = lim b → ∞ [ ∫ a b f ( x ) d x ] {\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\left[\int _{a}^{b}f(x)\,dx\right]}
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.
Definite integrals involving rational or irrational expressions
∫ 0 ∞ d x 1 + x p = π / p sin ( π / p ) for ℜ ( p ) > 1 {\displaystyle \int _{0}^{\infty }{\frac {dx}{1+x^{p}}}={\frac {\pi /p}{\sin(\pi /p)}}\quad {\text{for }}\Re (p)>1}
∫ 0 ∞ d x a p + x p = π p a p − 1 sin ( π p ) for ℜ ( p ) > 1 , ℜ ( a ) > 0 {\displaystyle \int _{0}^{\infty }{\frac {dx}{a^{p}+x^{p}}}={\frac {\pi }{pa^{p-1}\sin {({\frac {\pi }{p}}})}}\quad {\text{for }}\Re (p)>1{\text{, }}\Re (a)>0}
∫ 0 ∞ x p − 1 d x 1 + x = π sin ( p π ) for 0 < p < 1 {\displaystyle \int _{0}^{\infty }{\frac {x^{p-1}dx}{1+x}}={\frac {\pi }{\sin(p\pi )}}\quad {\text{for }}0<p<1}
∫ 0 ∞ x m d x x n + a n = π a m − n + 1 n sin ( m + 1 n π ) for 0 < m + 1 < n {\displaystyle \int _{0}^{\infty }{\frac {x^{m}dx}{x^{n}+a^{n}}}={\frac {\pi a^{m-n+1}}{n\sin \left({\dfrac {m+1}{n}}\pi \right)}}\quad {\text{for }}0<m+1<n}
∫ 0 ∞ x m d x 1 + 2 x cos β + x 2 = π sin ( m π ) ⋅ sin ( m β ) sin ( β ) {\displaystyle \int _{0}^{\infty }{\frac {x^{m}dx}{1+2x\cos \beta +x^{2}}}={\frac {\pi }{\sin(m\pi )}}\cdot {\frac {\sin(m\beta )}{\sin(\beta )}}}
∫ 0 a d x a 2 − x 2 = π 2 {\displaystyle \int _{0}^{a}{\frac {dx}{\sqrt {a^{2}-x^{2}}}}={\frac {\pi }{2}}}
∫ 0 a a 2 − x 2 d x = π a 2 4 {\displaystyle \int _{0}^{a}{\sqrt {a^{2}-x^{2}}}dx={\frac {\pi a^{2}}{4}}}
∫ 0 a x m ( a n − x n ) p d x = a m + 1 + n p Γ ( m + 1 n ) Γ ( p + 1 ) n Γ ( m + 1 n + p + 1 ) {\displaystyle \int _{0}^{a}x^{m}(a^{n}-x^{n})^{p}\,dx={\frac {a^{m+1+np}\Gamma \left({\dfrac {m+1}{n}}\right)\Gamma (p+1)}{n\Gamma \left({\dfrac {m+1}{n}}+p+1\right)}}}
∫ 0 ∞ x m d x ( x n + a n ) r = ( − 1 ) r − 1 π a m + 1 − n r Γ ( m + 1 n ) n sin ( m + 1 n π ) ( r − 1 ) ! Γ ( m + 1 n − r + 1 ) for n ( r − 2 ) < m + 1 < n r {\displaystyle \int _{0}^{\infty }{\frac {x^{m}dx}{({x^{n}+a^{n})}^{r}}}={\frac {(-1)^{r-1}\pi a^{m+1-nr}\Gamma \left({\dfrac {m+1}{n}}\right)}{n\sin \left({\dfrac {m+1}{n}}\pi \right)(r-1)!\,\Gamma \left({\dfrac {m+1}{n}}-r+1\right)}}\quad {\text{for }}n(r-2)<m+1<nr}
Definite integrals involving trigonometric functions
∫ 0 π sin ( m x ) sin ( n x ) d x = { 0 if m ≠ n π 2 if m = n for m , n positive integers {\displaystyle \int _{0}^{\pi }\sin(mx)\sin(nx)dx={\begin{cases}0&{\text{if }}m\neq n\\\\{\dfrac {\pi }{2}}&{\text{if }}m=n\end{cases}}\quad {\text{for }}m,n{\text{ positive integers}}}
∫ 0 π cos ( m x ) cos ( n x ) d x = { 0 if m ≠ n π 2 if m = n for m , n positive integers {\displaystyle \int _{0}^{\pi }\cos(mx)\cos(nx)dx={\begin{cases}0&{\text{if }}m\neq n\\\\{\dfrac {\pi }{2}}&{\text{if }}m=n\end{cases}}\quad {\text{for }}m,n{\text{ positive integers}}}
∫ 0 π sin ( m x ) cos ( n x ) d x = { 0 if m + n even 2 m m 2 − n 2 if m + n odd for m , n integers . {\displaystyle \int _{0}^{\pi }\sin(mx)\cos(nx)dx={\begin{cases}0&{\text{if }}m+n{\text{ even}}\\\\{\dfrac {2m}{m^{2}-n^{2}}}&{\text{if }}m+n{\text{ odd}}\end{cases}}\quad {\text{for }}m,n{\text{ integers}}.}
∫ 0 π 2 sin 2 ( x ) d x = ∫ 0 π 2 cos 2 ( x ) d x = π 4 {\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{2}(x)dx=\int _{0}^{\frac {\pi }{2}}\cos ^{2}(x)dx={\frac {\pi }{4}}}
∫ 0 π 2 sin 2 m ( x ) d x = ∫ 0 π 2 cos 2 m ( x ) d x = 1 × 3 × 5 × ⋯ × ( 2 m − 1 ) 2 × 4 × 6 × ⋯ × 2 m ⋅ π 2 for m = 1 , 2 , 3 … {\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{2m}(x)dx=\int _{0}^{\frac {\pi }{2}}\cos ^{2m}(x)dx={\frac {1\times 3\times 5\times \cdots \times (2m-1)}{2\times 4\times 6\times \cdots \times 2m}}\cdot {\frac {\pi }{2}}\quad {\text{for }}m=1,2,3\ldots }
∫ 0 x sin 2 m ( t ) d t = ( 2 m − 1 ) ! ! ( 2 m ) ! ! ( x − sin ( x ) cos ( x ) ( 1 + ∑ k = 1 ∞ sin 2 k ( x ) ( 2 k ) ! ! ( 2 k + 1 ) ! ! ) ) for m = 1 , 2 , 3 … {\displaystyle \int _{0}^{x}\sin ^{2m}(t)dt={\frac {(2m-1)!!}{(2m)!!}}{\Biggl (}x-\sin(x)\cos(x){\Biggl (}1+\sum _{k=1}^{\infty }{\frac {\sin ^{2k}(x)(2k)!!}{(2k+1)!!}}{\Biggr )}{\Biggr )}\quad {\text{for }}m=1,2,3\ldots }
∫ 0 x cos 2 m ( t ) d t = ( 2 m − 1 ) ! ! ( 2 m ) ! ! ( x − sin ( x ) cos ( x ) ( 1 + ∑ k = 1 ∞ cos 2 k ( x ) ( 2 k ) ! ! ( 2 k + 1 ) ! ! ) ) for m = 1 , 2 , 3 … {\displaystyle \int _{0}^{x}\cos ^{2m}(t)dt={\frac {(2m-1)!!}{(2m)!!}}{\Biggl (}x-\sin(x)\cos(x){\Biggl (}1+\sum _{k=1}^{\infty }{\frac {\cos ^{2k}(x)(2k)!!}{(2k+1)!!}}{\Biggr )}{\Biggr )}\quad {\text{for }}m=1,2,3\ldots }
∫ 0 π 2 sin 2 m + 1 ( x ) d x = ∫ 0 π 2 cos 2 m + 1 ( x ) d x = 2 × 4 × 6 × ⋯ × 2 m 1 × 3 × 5 × ⋯ × ( 2 m + 1 ) for m = 1 , 2 , 3 … {\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{2m+1}(x)dx=\int _{0}^{\frac {\pi }{2}}\cos ^{2m+1}(x)dx={\frac {2\times 4\times 6\times \cdots \times 2m}{1\times 3\times 5\times \cdots \times (2m+1)}}\quad {\text{for }}m=1,2,3\ldots }
∫ 0 π 2 sin 2 p − 1 ( x ) cos 2 q − 1 ( x ) d x = Γ ( p ) Γ ( q ) 2 Γ ( p + q ) = 1 2 B ( p , q ) {\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{2p-1}(x)\cos ^{2q-1}(x)dx={\frac {\Gamma (p)\Gamma (q)}{2\Gamma (p+q)}}={\frac {1}{2}}{\text{B}}(p,q)}
∫ 0 ∞ sin ( p x ) x d x = { π 2 if p > 0 0 if p = 0 − π 2 if p < 0 {\displaystyle \int _{0}^{\infty }{\frac {\sin(px)}{x}}dx={\begin{cases}{\dfrac {\pi }{2}}&{\text{if }}p>0\\\\0&{\text{if }}p=0\\\\-{\dfrac {\pi }{2}}&{\text{if }}p<0\end{cases}}} (see Dirichlet integral)
∫ 0 ∞ sin p x cos q x x d x = { 0 if q > p > 0 π 2 if 0 < q < p π 4 if p = q > 0 {\displaystyle \int _{0}^{\infty }{\frac {\sin px\cos qx}{x}}\ dx={\begin{cases}0&{\text{ if }}q>p>0\\\\{\dfrac {\pi }{2}}&{\text{ if }}0<q<p\\\\{\dfrac {\pi }{4}}&{\text{ if }}p=q>0\end{cases}}}
∫ 0 ∞ sin p x sin q x x 2 d x = { π p 2 if 0 < p ≤ q π q 2 if 0 < q ≤ p {\displaystyle \int _{0}^{\infty }{\frac {\sin px\sin qx}{x^{2}}}\ dx={\begin{cases}{\dfrac {\pi p}{2}}&{\text{ if }}0<p\leq q\\\\{\dfrac {\pi q}{2}}&{\text{ if }}0<q\leq p\end{cases}}}
∫ 0 ∞ sin 2 p x x 2 d x = π p 2 {\displaystyle \int _{0}^{\infty }{\frac {\sin ^{2}px}{x^{2}}}\ dx={\frac {\pi p}{2}}}
∫ 0 ∞ 1 − cos p x x 2 d x = π p 2 {\displaystyle \int _{0}^{\infty }{\frac {1-\cos px}{x^{2}}}\ dx={\frac {\pi p}{2}}}
∫ 0 ∞ cos p x − cos q x x d x = ln q p {\displaystyle \int _{0}^{\infty }{\frac {\cos px-\cos qx}{x}}\ dx=\ln {\frac {q}{p}}}
∫ 0 ∞ cos p x − cos q x x 2 d x = π ( q − p ) 2 {\displaystyle \int _{0}^{\infty }{\frac {\cos px-\cos qx}{x^{2}}}\ dx={\frac {\pi (q-p)}{2}}}
∫ 0 ∞ cos m x x 2 + a 2 d x = π 2 a e − m a {\displaystyle \int _{0}^{\infty }{\frac {\cos mx}{x^{2}+a^{2}}}\ dx={\frac {\pi }{2a}}e^{-ma}}
∫ 0 ∞ x sin m x x 2 + a 2 d x = π 2 e − m a {\displaystyle \int _{0}^{\infty }{\frac {x\sin mx}{x^{2}+a^{2}}}\ dx={\frac {\pi }{2}}e^{-ma}}
∫ 0 ∞ sin m x x ( x 2 + a 2 ) d x = π 2 a 2 ( 1 − e − m a ) {\displaystyle \int _{0}^{\infty }{\frac {\sin mx}{x(x^{2}+a^{2})}}\ dx={\frac {\pi }{2a^{2}}}\left(1-e^{-ma}\right)}
∫ 0 2 π d x a + b sin x = 2 π a 2 − b 2 {\displaystyle \int _{0}^{2\pi }{\frac {dx}{a+b\sin x}}={\frac {2\pi }{\sqrt {a^{2}-b^{2}}}}}
∫ 0 2 π d x a + b cos x