List of definite integrals

:$\backslash int\_0^\backslash infty\; \backslash frac\{dx\}\{1+x^p\}=\backslash frac\{\backslash pi/p\}\{\backslash sin(\backslash pi/p)\}\backslash quad\; \backslash text\{for\; \}\; \backslash Re(p)>1$

:

:

:$\backslash int\_0^\backslash infty\; \backslash frac\{x^m\; dx\}\{1+2x\backslash cos\backslash beta+x^\{2\}\}=\backslash frac\{\backslash pi\}\{\backslash sin(m\backslash pi)\}\backslash cdot\backslash frac\{\backslash sin(m\backslash beta)\}\{\backslash sin(\backslash beta)\}$

:$\backslash int\_0^a\; \backslash frac\{dx\}\{\backslash sqrt\{a^2-x^2\}\}=\backslash frac\{\backslash pi\}\{2\}$

:$\backslash int\_0^a\; \backslash sqrt\{a^2-x^2\}dx=\backslash frac\{\backslash pi\; a^2\}\{4\}$

:$\backslash int\_0^a\; x^m\; (a^n-x^n)^p\backslash ,dx=\backslash frac\{a^\{m+1+np\}\backslash Gamma\backslash left(\backslash dfrac\{m+1\}\{n\}\backslash right)\backslash Gamma(p+1)\}\{n\backslash Gamma\; \backslash left(\backslash dfrac\{m+1\}\{n\}+p+1\backslash right)\}$

:

:$\backslash int\_0^\backslash pi\; \backslash sin(mx)\backslash sin(nx)dx=\backslash begin\{cases\}$

0 & \text{if } m\ne n \\ \\

\dfrac{\pi}{2} & \text{if } m=n

\end{cases}

\quad\text{for } m,n \text{ positive integers}

:$\backslash int\_0^\backslash pi\; \backslash cos(mx)\backslash cos(nx)dx=\backslash begin\{cases\}$

0 & \text{if } m\ne n \\ \\

\dfrac{\pi}{2} & \text{if } m=n

\end{cases}

\quad\text{for } m,n \text{ positive integers}

:$\backslash int\_0^\backslash pi\; \backslash sin(mx)\backslash cos(nx)dx=\backslash begin\{cases\}$

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}.

:$\backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\backslash sin^2(x)dx=\backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\backslash cos^2(x)dx=\backslash frac\{\backslash pi\}\{4\}$

:$\backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\backslash sin^\{2m\}(x)dx=\backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\backslash cos^\{2m\}(x)dx\; =\; \backslash frac\{1\backslash times3\backslash times5\backslash times\backslash cdots\backslash times(2m-1)\}\{2\backslash times4\backslash times6\backslash times\backslash cdots\backslash times2m\}\backslash cdot\backslash frac\{\backslash pi\}\{2\}\backslash quad\backslash text\{for\; \}m=1,2,3\backslash ldots$

:$\backslash int\_0^\{x\}\backslash sin^\{2m\}(t)dt=\backslash frac\{(2m-1)!!\}\{(2m)!!\}\backslash Biggl(x-\backslash sin(x)\backslash cos(x)\backslash Biggl(1+\backslash sum\_\{k=1\}^\backslash infty\backslash frac\{\backslash sin^\{2k\}(x)(2k)!!\}\{(2k+1)!!\}\backslash Biggr)\backslash Biggr)\backslash quad\backslash text\{for\; \}m=1,2,3\backslash ldots$

:$\backslash int\_0^\{x\}\backslash cos^\{2m\}(t)dt=\backslash frac\{(2m-1)!!\}\{(2m)!!\}\backslash Biggl(x-\backslash sin(x)\backslash cos(x)\backslash Biggl(1+\backslash sum\_\{k=1\}^\backslash infty\backslash frac\{\backslash cos^\{2k\}(x)(2k)!!\}\{(2k+1)!!\}\backslash Biggr)\backslash Biggr)\backslash quad\backslash text\{for\; \}m=1,2,3\backslash ldots$

:$\backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\backslash sin^\{2m+1\}(x)dx=\backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\backslash cos^\{2m+1\}(x)dx\; =\; \backslash frac\{2\backslash times4\backslash times6\backslash times\backslash cdots\backslash times2m\}\{1\backslash times3\backslash times5\backslash times\backslash cdots\backslash times(2m+1)\}\backslash quad\backslash text\{for\; \}m=1,2,3\backslash ldots$

:$\backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\backslash sin^\{2p-1\}(x)\backslash cos^\{2q-1\}(x)dx\; =\; \backslash frac\{\backslash Gamma(p)\backslash Gamma(q)\}\{2\backslash Gamma(p+q)\}=\backslash frac\{1\}\{2\}\; \backslash text\{B\}(p,q)$

:$\backslash int\_0^\backslash infty\; \backslash frac\{\backslash sin(px)\}\{x\}dx=\backslash begin\{cases\}$

\dfrac{\pi}{2} & \text{if } p>0 \\

\\

0 & \text{if } p=0 \\

\\

-\dfrac{\pi}{2} & \text {if } p<0

\end{cases} (see Dirichlet integral)

: $\backslash int\_\{0\}^\{\backslash infty\; \}\backslash frac\{\backslash sin\; px\backslash cos\; qx\}\{x\}\backslash \; dx=\backslash begin\{cases\}$

0 & \text{ if } q>p>0 \\

\\

\dfrac{\pi}{2}& \text{ if } 0

\\

\dfrac{\pi}{4} & \text{ if } p=q>0

\end{cases}

: $\backslash int\_\{0\}^\{\backslash infty\; \}\backslash frac\{\backslash sin\; px\; \backslash sin\; qx\}\{x^\{2\}\}\backslash \; dx=\backslash begin\{cases\}$

\dfrac{\pi p}{2}& \text{ if } 0

\\

\dfrac{\pi q}{2} & \text{ if } 0

\end{cases}

: $\backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash sin\; ^\{2\}px\}\{x^\{2\}\}\backslash \; dx=\backslash frac\{\backslash pi\; p\}\{2\}$

: $\backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{1-\backslash cos\; px\}\{x^\{2\}\}\backslash \; dx=\backslash frac\{\backslash pi\; p\}\{2\}$

: $\backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash cos\; px\; -\; \backslash cos\; qx\}\{x\}\backslash \; dx=\; \backslash ln\; \backslash frac\; \{q\}\{p\}$

: $\backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash cos\; px\; -\; \backslash cos\; qx\}\{x^\{2\}\}\backslash \; dx=\backslash frac\{\backslash pi\; (q-p)\}\{2\}$

: $\backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash cos\; mx\}\{x^\{2\}+a^\{2\}\}\backslash \; dx=\backslash frac\{\backslash pi\}\{2a\}e^\{-ma\}$

: $\backslash int\_0^\backslash infty\; \backslash frac\{x\; \backslash sin\; mx\}\{x^2+a^2\}\backslash \; dx=\backslash frac\{\backslash pi\}\{2\}e^\{-ma\}$

: $\backslash int\_0^\backslash infty\; \backslash frac\{\; \backslash sin\; mx\}\{x(x^2+a^2)\}\backslash \; dx=\backslash frac\{\backslash pi\}\{2a^2\}\backslash left(1-e^\{-ma\}\backslash right)$

: $\backslash int\_0^\{2\backslash pi\}\; \backslash frac\{dx\}\{a+b\backslash sin\; x\}=\backslash frac\{2\backslash pi\}\{\backslash sqrt\{a^2-b^2\}\}$

: $\backslash int\_0^\{2\backslash pi\}\; \backslash frac\{dx\}\{a+b\backslash cos\; x\}=\backslash frac\{2\backslash pi\}\{\backslash sqrt\{a^2-b^2\}\}$

: $\backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\; \backslash frac\{dx\}\{a+b\backslash cos\; x\}=\backslash frac\{\backslash cos^\{-1\}\; \backslash left(\backslash dfrac\{b\}\{a\}\backslash right)\; \}\{\backslash sqrt\{a^2-b^2\}\}$

: $\backslash int\_0^\{2\backslash pi\}\; \backslash frac\{dx\}\{(a+b\backslash sin\; x)^2\}=\backslash int\_0^\{2\backslash pi\}\; \backslash frac\{dx\}\{(a+b\backslash cos\; x)^2\}=\backslash frac\{2\backslash pi\; a\}\{(a^2-b^2)^\{3/2\}\}$

:

: $\backslash int\_0^\{\backslash pi\}\; \backslash frac\{x\; \backslash sin\; x\backslash \; dx\}\{1-2a\backslash cos\; x\; +a^2\}=\backslash begin\{cases\}$

\dfrac{\pi}{a}\ln\left|1+a\right| & \text{if } |a|<1 \\ \\

\dfrac{\pi}{a} \ln\left|1+\dfrac{1}{a}\right| & \text{if } |a|>1

\end{cases}

:

: $\backslash int\_0^\backslash infty\; \backslash sin\; ax^2\backslash \; dx=\backslash int\_0^\backslash infty\; \backslash cos\; ax^2=\; \backslash frac\{1\}\{2\}\backslash sqrt\; \backslash frac\{\backslash pi\}\{2a\}$

: $\backslash int\_0^\backslash infty\; \backslash sin\; ax^n=\backslash frac\{1\}\{na^\{1/n\}\}\backslash Gamma\backslash left(\backslash frac\{1\}\{n\}\backslash right)\backslash sin\backslash frac\{\backslash pi\}\{2n\}\; \backslash quad\; \backslash text\{for\; \}\; n>1$

: $\backslash int\_0^\backslash infty\; \backslash cos\; ax^n=\backslash frac\{1\}\{na^\{1/n\}\}\backslash Gamma\backslash left(\backslash frac\{1\}\{n\}\backslash right)\backslash cos\backslash frac\{\backslash pi\}\{2n\}\; \backslash quad\; \backslash text\{for\; \}\; n>1$

: $\backslash int\_0^\backslash infty\; \backslash frac\{\backslash sin\; x\}\{\backslash sqrt\; x\}\backslash \; dx=\backslash int\_0^\backslash infty\; \backslash frac\{\backslash cos\; x\}\{\backslash sqrt\; x\}\backslash \; dx=\backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}$

:

:

: $\backslash int\_0^\backslash infty\; \backslash sin\; ax^2\backslash cos\; 2bx\backslash \; dx=\backslash frac\{1\}\{2\}\backslash sqrt\{\backslash frac\{\backslash pi\}\{2a\}\}\backslash left(\backslash cos\; \backslash frac\{b^2\}\{a\}-\backslash sin\backslash frac\{b^2\}\{a\}\backslash right)$

: $\backslash int\_0^\backslash infty\; \backslash cos\; ax^2\backslash cos\; 2bx\backslash \; dx=\backslash frac\{1\}\{2\}\backslash sqrt\{\backslash frac\{\backslash pi\}\{2a\}\}\backslash left(\backslash cos\; \backslash frac\{b^2\}\{a\}+\backslash sin\backslash frac\{b^2\}\{a\}\backslash right)$

: $\backslash int\_0^\backslash infty\; \backslash sqrt\{x\}\backslash ,e^\{-x\}\backslash ,dx\; =\; \backslash frac\{1\}\{2\}\backslash sqrt\; \backslash pi$ (see also Gamma function)

: $\backslash int\_0^\backslash infty\; e^\{-ax\}\backslash cos\; bx\; \backslash ,\; dx=\backslash frac\{a\}\{a^2+b^2\}$

: $\backslash int\_0^\backslash infty\; e^\{-ax\}\backslash sin\; bx\; \backslash ,\; dx=\backslash frac\{b\}\{a^\{2\}+b^\{2\}\}$

: $\backslash int\_0^\backslash infty\; \backslash frac\; \{\{\}e^\{-ax\}\backslash sin\; bx\}\{x\}\; \backslash ,\; dx=\backslash tan^\{-1\}\backslash frac\{b\}\{a\}$

: $\backslash int\_0^\backslash infty\; \backslash frac\; \{e^\{-ax\}-e^\{-bx\}\}\{x\}\; \backslash ,\; dx=\backslash ln\; \backslash frac\{b\}\{a\}$

: $\backslash int\_0^\backslash infty\; \backslash frac\; \{e^\{-ax\}-\backslash cos(bx)\}\{x\}\; \backslash ,\; dx=\backslash ln\; \backslash frac\{b\}\{a\}$

: $\backslash int\_0^\backslash infty\; e^\{-a\; x^2\}\backslash ,dx\; =\; \backslash frac\{1\}\{2\}\; \backslash sqrt\; \backslash frac\; \{\backslash pi\}\; \{a\}\; \backslash quad\; \backslash text\{for\; \}\; a>0$ (the Gaussian integral)

: $\backslash int\_0^\backslash infty\; \{e^\{-ax^\{2\}\}\}\backslash cos\; bx\backslash ,\; dx=\backslash frac\; \{1\}\{2\}\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{a\}\}e^\backslash left(\backslash frac\{-b^2\}\{4a\}\backslash right)$

: $\backslash int\_0^\backslash infty\; e^\{-(ax^\{2\}+bx+c)\}\backslash ,\; dx=\backslash frac\{1\}\{2\}\backslash sqrt\{\backslash frac\{\backslash pi\}\{a\}\}e^\backslash left(\backslash frac\{b^2-4ac\}\{4a\}\backslash right)\backslash cdot\; \backslash operatorname\{erfc\}\; \backslash frac\{b\}\{2\backslash sqrt\{a\}\},\backslash text\{\; where\; \}\backslash operatorname\{erfc\}(p)=\backslash frac\{2\}\{\backslash sqrt\{\backslash pi\}\}\backslash int\_p^\backslash infty\; e^\{-x^\{2\}\}\backslash ,\; dx$

: $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{-(ax^\{2\}+bx+c)\}\backslash \; dx=\backslash sqrt\; \{\backslash frac\{\backslash pi\}\{a\}\}e^\backslash left(\backslash frac\{b^2-4ac\}\{4a\}\backslash right)$

: $\backslash int\_0^\backslash infty\; x^\{n\}e^\{-ax\}\backslash \; dx=\backslash frac\{\backslash Gamma\; (n+1)\}\{a^\{n+1\}\}$

: $\backslash int\_0^\backslash infty\{x^2\; e^\{-a\; x^2\}\backslash ,dx\}\; =\; \backslash frac\{1\}\{4\}\; \backslash sqrt\; \backslash frac\; \{\backslash pi\}\; \{a^3\}\; \backslash quad\; \backslash text\{for\; \}\; a>0$

: $\backslash int\_0^\backslash infty\; x^\{2n\}\; e^\{-a\; x^2\}\backslash ,dx$

= \frac{2n-1}{2a} \int_0^\infty x^{2(n-1)} e^{-a x^2}\,dx

= \frac{(2n-1)!!}{2^{n+1}} \sqrt{\frac{\pi}{a^{2n+1}}}

= \frac{(2n)!}{n! 2^{2n+1}} \sqrt{\frac{\pi}{a^{2n+1}}} \quad \text{for } a>0\ ,\ n=1,2,3\ldots (where !! is the double factorial)

: $\backslash int\_0^\backslash infty\{x^3\; e^\{-a\; x^2\}\backslash ,dx\}\; =\; \backslash frac\{1\}\{2\; a^2\}\; \backslash quad\; \backslash text\{for\; \}\; a>0$

: $\backslash int\_0^\backslash infty\; x^\{2n+1\}\; e^\{-a\; x^2\}\backslash ,dx$

= \frac {n} {a} \int_0^\infty x^{2n-1} e^{-a x^2}\,dx

= \frac{n!}{2 a^{n+1}} \quad \text{for } a>0\ ,\ n=0,1,2\ldots

: $\backslash int\_0^\backslash infty\; x^m\; e^\{-ax^2\}\backslash \; dx=\backslash frac\{\backslash Gamma\backslash left(\backslash dfrac\{m+1\}\{2\}\backslash right)\}\{2a^\backslash left(\backslash frac\{m+1\}\{2\}\backslash right)\}$

: $\backslash int\_0^\backslash infty\; e^\{\backslash left(-ax^2-\backslash frac\{b\}\{x^2\}\backslash right)\}\backslash \; dx=\backslash frac\{1\}\{2\}\; \backslash sqrt\; \backslash frac\{\backslash pi\}\{a\}e^\{-2\; \backslash sqrt\{ab\}\}$

: $\backslash int\_0^\backslash infty\; \backslash frac\; \{x\}\{e^x-1\}\backslash \; dx=\backslash zeta\; (2)=\; \backslash frac\; \{\backslash pi^2\}\{6\}$

: $\backslash int\_0^\backslash infty\; \backslash frac\; \{x^\{n-1\}\}\{e^x-1\}\backslash \; dx=\backslash Gamma\; (n)\backslash zeta\; (n)$

: $\backslash int\_0^\backslash infty\; \backslash frac\; \{x\}\{e^x+1\}\backslash \; dx=\backslash frac\{1\}\{1^2\}-\backslash frac\{1\}\{2^2\}+\backslash frac\{1\}\{3^2\}-\backslash frac\{1\}\{4^2\}+\backslash dots=\backslash frac\{\backslash pi^2\}\{12\}$

: $\backslash int\_0^\backslash infty\; \backslash frac\; \{x^n\}\{e^x+1\}\backslash \; dx=\; n!\; \backslash cdot\; \backslash left(\; \backslash frac\{2^n-1\}\{2^n\}\; \backslash right)\; \backslash zeta(n+1)$

: $\backslash int\_0^\backslash infty\; \backslash frac\; \{\backslash sin\; mx\}\{e^\{2\backslash pi\; x\}-1\}\backslash \; dx=\backslash frac\{1\}\{4\}\; \backslash coth\backslash frac\{m\}\{2\}-\; \backslash frac\{1\}\{2m\}$

: $\backslash int\_0^\backslash infty\; \backslash left(\backslash frac\; \{1\}\{1+x\}-\; e^\{-x\}\backslash right)\backslash \; \backslash frac\{dx\}\{x\}=\backslash gamma$ (where $\backslash gamma$ is Euler–Mascheroni constant)

: $\backslash int\_0^\backslash infty\; \backslash frac\; \{e^\{-x^2\}-e^\{-x\}\}\{x\}\backslash \; dx=\backslash frac\{\backslash gamma\}\{2\}$

: $\backslash int\_0^\backslash infty\; \backslash left(\backslash frac\; \{1\}\{e^x-1\}-\backslash frac\{e^\{-x\}\}\{x\}\backslash right)\backslash \; dx=\backslash gamma$

: $\backslash int\_0^\backslash infty\; \backslash frac\; \{e^\{-ax\}-e^\{-bx\}\}\{x\; \backslash sec\; px\}\backslash \; dx=\backslash frac\{1\}\{2\}\; \backslash ln\backslash frac\{b^2+p^2\}\{a^2+p^2\}$

: $\backslash int\_0^\backslash infty\; \backslash frac\; \{e^\{-ax\}-e^\{-bx\}\}\{x\; \backslash csc\; px\}\backslash \; dx=\backslash tan^\{-1\}\backslash frac\{b\}\{p\}-\backslash tan^\{-1\}\backslash frac\{a\}\{p\}$

: $\backslash int\_0^\backslash infty\; \backslash frac\; \{e^\{-ax\}(1-\backslash cos\; x)\}\{x^2\}\backslash \; dx=\backslash cot^\{-1\}\; a-\backslash frac\{a\}\{2\}\backslash ln\backslash left|\backslash frac\{a^2+1\}\{a^2\}\backslash right|$

: $\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-x^2\}\backslash ,dx=\backslash sqrt\{\backslash pi\}$

: $\backslash int\_\{-\backslash infty\}^\backslash infty\; x^\{2(n+1)\}e^\{-\backslash frac12\; x^2\}\backslash ,dx=\backslash frac\{(2n+1)!\}\{2^n\; n!\}\backslash sqrt\{2\; \backslash pi\}\; \backslash quad\backslash text\{for\; \}\; n=0,1,2,\backslash ldots$

: $\backslash int\_0^1\; x^m\; (\backslash ln\; x)^n\; \backslash ,\; dx=\backslash frac\{(-1)^n\; n!\}\{(m+1)^\{n+1\}\}\; \backslash quad\backslash text\{for\; \}\; m>-1,\; n=0,1,2,\backslash ldots$

:

: $\backslash int\_0^1\; \backslash frac\{\backslash ln\; x\}\{1+x\}\backslash ,\; dx=\; -\backslash frac\{\backslash pi^2\}\{12\}$

: $\backslash int\_0^1\; \backslash frac\{\backslash ln\; x\}\{1-x\}\backslash ,\; dx=\; -\backslash frac\{\backslash pi^2\}\{6\}$

: $\backslash int\_0^1\; \backslash frac\{\backslash ln\; (1+x)\}\{x\}\backslash ,\; dx=\; \backslash frac\{\backslash pi^2\}\{12\}$

: $\backslash int\_0^1\; \backslash frac\{\backslash ln\; (1-x)\}\{x\}\backslash ,\; dx=\; -\backslash frac\{\backslash pi^2\}\{6\}$

: $\backslash int\_0^\backslash infty\; \backslash frac\{\backslash ln(a^\{2\}+x^\{2\})\}\{b^\{2\}+x^\{2\}\}\backslash \; dx\; =\; \backslash frac\{\backslash pi\}\{b\}\; \backslash ln\; (a+b)\backslash quad\backslash text\{for\; \}\; a,b>0$

: $\backslash int\_0^\backslash infty\backslash frac\{\backslash ln\; x\}\{x^2+a^2\}\backslash \; dx\; =\; \backslash frac\{\backslash pi\; \backslash ln\; a\}\{2a\}\backslash quad\backslash text\{for\; \}\; a>0$

$$\backslash int\_0^\backslash infty\; \backslash frac\{\backslash sin\; ax\}\{\backslash sinh\; bx\}\backslash \; dx=\backslash frac\; \{\backslash pi\}\{2b\}\backslash tanh\; \backslash frac\{a\; \backslash pi\}\{2b\}$$

$$\backslash int\_0^\backslash infty\; \backslash frac\{\backslash cos\; ax\}\{\backslash cosh\; bx\}\backslash \; dx=\backslash frac\; \{\backslash pi\}\{2b\}\backslash cdot\backslash frac\{1\}\{\backslash cosh\; \backslash frac\{a\; \backslash pi\}\{2b\}\}$$

$$\backslash int\_0^\backslash infty\; \backslash frac\{x\}\{\backslash sinh\; ax\}\backslash \; dx=\backslash frac\{\backslash pi^2\}\{4a^2\}$$

$$\backslash int\_0^\backslash infty\; \backslash frac\{x^\{2n+1\}\}\{\backslash sinh\; ax\}\backslash \; dx=\; c\_\{2n+1\}\; \backslash left(\; \backslash frac\{\backslash pi\}\{a\}\; \backslash right)^\{2(n+1)\},\; \backslash quad\; c\_\{2n+1\}\; =\; \backslash frac\{(-1)^n\}\{2\}\; \backslash left(\backslash frac\{1\}\{2\}-\; \backslash sum\_\{k=0\}^\{n-1\}\; (-1)^k\; \{2n+1\; \backslash choose\; 2k+1\}\; c\_\{2k+1\}\; \backslash right),\; \backslash quad\; c\_1\; =\; \backslash frac\{1\}\{4\}$$

$$\backslash int\_\{0\}^\backslash infty\; \backslash frac\{1\}\{\backslash cosh\; ax\}\backslash \; dx\; =\; \backslash frac\{\backslash pi\}\{2a\}$$

$$\backslash int\_0^\backslash infty\; \backslash frac\{x^\{2n\}\}\{\backslash cosh\; ax\}\backslash \; dx=\; d\_\{2n\}\; \backslash left(\; \backslash frac\{\backslash pi\}\{a\}\; \backslash right)^\{2n+1\},\; \backslash quad\; d\_\{2n\}\; =\; \backslash frac\{(-1)^n\}\{2\}\; \backslash left(\backslash frac\{1\}\{4^n\}-\; \backslash sum\_\{k=0\}^\{n-1\}\; (-1)^k\; \{2n\; \backslash choose\; 2k\}\; d\_\{2k\}\; \backslash right),\; \backslash quad\; d\_0\; =\; \backslash frac\{1\}\{2\}$$

$$\backslash int\_\{0\}^\{\backslash infty\; \}\backslash frac\{f(ax)-f(bx)\}\{x\}\backslash \; dx=\backslash left(\backslash lim\_\{x\; \backslash to\; 0\}f(x)-\backslash lim\_\{x\; \backslash to\; \backslash infty\}f(x)\backslash right)\; \backslash ln\; \backslash left(\backslash frac\{b\}\{a\}\; \backslash right)$$ holds if the integral exists and $f\text{'}(x)$ is continuous.

{{Portal|Mathematics}}

• List of integrals
• Indefinite sum
• Gamma function
• List of limits

• {{cite journal |author-first1=Robert |author-last1=Reynolds |author-first2=Allan |author-last2=Stauffer |title = Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions |journal = Mathematics |year = 2020 |volume=8 |number = 687 |page=687 |doi = 10.3390/math8050687 |doi-access = free }}
• {{cite journal |author-first1=Robert |author-last1=Reynolds |author-first2=Allan |author-last2=Stauffer |title = A Definite Integral Involving the Logarithmic Function in Terms of the Lerch Function |journal = Mathematics |year = 2019 |volume=7 |number = 1148 |page=1148 |doi = 10.3390/math7121148 |doi-access = free }}
• {{cite journal |author-first1=Robert |author-last1=Reynolds |author-first2=Allan |author-last2=Stauffer |title = Definite Integral of Arctangent and Polylogarithmic Functions Expressed as a Series |journal = Mathematics |year = 2019 |volume=7 |number = 1099 |page=1099 |doi = 10.3390/math7111099 |doi-access = free }}
• {{cite journal |author-first1=Anton |author-last1=Winckler |title = Eigenschaften Einiger Bestimmten Integrale |journal = Hof, K.K., Ed. |year = 1861 }}
• {{cite book |author-first1=Murray R. |author-last1=Spiegel |author-first2=Seymour |author-last2=Lipschutz |author-first3=John |author-last3=Liu |title=Mathematical handbook of formulas and tables |year=2009 |publisher=McGraw-Hill |isbn=978-0071548557 |edition=3rd}}
• {{cite book |author-last=Zwillinger |author-first=Daniel |title=CRC standard mathematical tables and formulae |year=2003 |publisher=CRC Press |isbn=978-143983548-7 |edition=32nd}}
• {{AS ref}}

{{Lists of integrals}}

{{DEFAULTSORT:Definite integrals}}

definite integrals