$${\int{\sin (x) \ dx} = -\cos(x) + C}$$
$${\int{\cos (x) \ dx} = \sin(x) + C}$$
$${\int{\sec (x) \ dx} = \ln |\sec(x) + \tan(x)| + C}$$
$${\int{\tan (x) \ dx} = \ln |\sec(x)| + C}$$
$${\int{\cot (x) \ dx} = \ln|\sin(x)| + C}$$
$${\int{\csc (x) \ dx} = \ln|\csc(x) - \cot(x)| + C}$$
$${\int{\csc (x) \ dx} = - \ln|\csc(x) + \cot(x)| + C}$$
$${\int{\sec^2 (x) \ dx} = \tan(x) + C}$$
$${\int{\csc^2 (x) \ dx} = -\cot(x) + C}$$
$${\int{\sec(x) \tan(x) \ dx} = \sec(x) + C}$$
$${\int{\sec^3 (x) \ dx} = \frac{1}{2} \sec(x) \tan(x) + \frac{1}{2} \ln |\sec(x) + \tan(x)| + C}$$
$${\int{\csc^3 (x) \ dx} = -\frac{1}{2} \csc(x) \cot(x) + \frac{1}{2} \ln |\csc(x) - \cot(x)| + C}$$
$${\int{\tan^2 (x) \ dx} = \tan(x) - x + C}$$
$${\int{\cot^2 (x) \ dx} = -\cot(x) - x + C}$$
$${\int{\frac{1}{a^2 + x^2} \ dx} = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C}$$
$${\int{\frac{1}{\sqrt{a^2 - x^2}} \ dx} = \sin^{-1}(\frac{x}{a}) + C}$$
$${\int{\frac{1}{x \sqrt{x^2 - a^2}} \ dx} = \frac{1}{a} \sec^{-1}|\frac{x}{a}| + C}$$
$${\int{\frac{1}{x^2 - a^2} \ dx} = -\frac{1}{a} \tanh^{-1}\left(\frac{x}{a}\right) + C}$$
$${\int{\frac{1}{x^2 - a^2} \ dx} = \frac{1}{2a} \ln \left\lvert \frac{x-a}{x+a} \right \rvert + C}$$
$${\int{\frac{1}{a^2 - x^2} \ dx} = \frac{1}{2a} \ln \left\lvert \frac{a+x}{a-x} \right \rvert + C}$$
$${\int{\frac{1}{\sqrt{1 + x^2}} \ dx} = \sinh^{-1}(x) + C}$$
$${\int{\frac{1}{\sqrt{x^2 - 1}} \ dx} = \cosh^{-1}(x) + C}$$
$${\int{\frac{1}{1 - x^2} \ dx} = \tanh^{-1}(x) + C}$$
$${\int{\sinh(x) \ dx} = \cosh(x) + C}$$
$${\int{\cosh(x) \ dx} = \sinh(x) + C}$$
$${\int{\tanh(x) \ dx} = \ln |\cosh(x)| + C}$$
$${\int{\coth(x) \ dx} = \ln|\sinh(x)| + C}$$
$${\int{\DeclareMathOperator{\sech}{sech}\sech(x) \ dx} = \tan^{-1}(\sinh(x)) + C}$$
$${\int{\DeclareMathOperator{\sech}{sech}\sech(x) \ dx} = 2 \ \tan^{-1}(e^x) + C}$$
$${\int{\DeclareMathOperator{\csch}{csch}\csch(x) \ dx} = \ln|\coth(x) - \DeclareMathOperator{\csch}{csch}\csch(x)| + C}$$
$${\int{\tanh^2(x) \ dx} = x - \tanh(x) + C}$$
$${\int{\coth^2(x) \ dx} = x - \coth(x) + C}$$
$${\int{\DeclareMathOperator{\sech}{sech}\sech^2(x) \ dx} = \tanh(x) + C}$$
$${\int{\DeclareMathOperator{\csch}{csch}\csch^2(x) \ dx} = - \coth(x) + C}$$
$${\int{\DeclareMathOperator{\sech}{sech}\sech(x) \tanh(x) \ dx} = -\DeclareMathOperator{\sech}{sech}\sech(x) + C}$$
$${\int{\DeclareMathOperator{\csch}{csch}\csch(x) \coth(x) \ dx} = -\DeclareMathOperator{\csch}{csch}\csch(x) + C}$$
$${\int{\ln (x) \ dx} = x \ \ln(x) - x + C}$$
$${\int{\frac{1}{x} \ dx} = \ln(x) + C}$$
$${\int_{a}^{b} {f(x) \ dx} = b f(b) - a f(a) - \int_{f(a)}^{f(b)} {f^{-1}(x) \ dx}}$$
$${\int_{a}^{b} {f(x) \ dx} + \int_{f(a)}^{f(b)} {f^{-1}(x) \ dx} = b f(b) - a f(a)}$$
$${\int e^{-x^2} \ dx = \frac{\sqrt{\pi}}{2} \text{erf}(x) + C}$$
$${\int e^{x^2} \ dx = \frac{\sqrt{\pi}}{2} \text{erfi}(x) + C}$$
$${\int \frac{e^{x}}{x} \ dx = \text{Ei}(x) + C}$$
$${\int \frac{1}{\ln(x)} \ dx = \text{li}(x) + C}$$
$${\int \frac{\sin(x)}{x} \ dx = \text{Si}(x) + C}$$
$${\int \frac{\cos(x)}{x} \ dx = \text{Ci}(x) + C}$$
$${\int \sin\left(\frac{\pi}{2} x^2 \right) \ dx = \text{S}(x) + C}$$
$${\int \cos\left(\frac{\pi}{2} x^2 \right) \ dx = \text{C}(x) + C}$$
$${\int_{0}^{x} \sin\left( t^2 \right) \ dx = \text{S}(x)}$$
$${\int_{0}^{x} \cos\left( t^2 \right) \ dx = \text{C}(x)}$$
$${- \int \frac{\ln(1-x)}{x} \ dx = \text{Li}_2(x) + C }$$
$${\int \frac{\ln(x)}{1-x} \ dx = \text{Li}_2(1 - x) + C}$$
Gaussian Integral:
$${\int_{-\infty}^{\infty} e^{-x^2} \ dx = \sqrt{\pi}}$$
$${\int_{0}^{\infty} e^{-x^2} \ dx = \frac{\sqrt{\pi}}{2}}$$
Gamma Function:
$${\Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} \ dt}$$
$${\Gamma\left(\frac{1}{2}\right) = \int_{0}^{\infty} t^{-\frac{1}{2}} e^{-t} \ dt = \sqrt{\pi}}$$
Beta Function:
$${B(z_1,z_2) = \frac{\Gamma(z_1)\Gamma(z_2)}{\Gamma(z_1 + z_2)}}$$
$${B(z_1,z_2) = \int_{0}^{1} t^{z_1 - 1} (1 - t)^{z_2 - 1} \ dt }$$
$${B(z_1,z_2) = 2 \int_{0}^{\frac{\pi}{2}} (\sin t)^{2z_1 - 1} (\cos t)^{2z_2 - 1} \ dt }$$
$${B(z_1,z_2) = n \int_{0}^{1} t^{n z_1 - 1} (1 - t^n)^{z_2 - 1} \ dt }$$
$${B(z_1,z_2) = \int_{0}^{\infty} \frac{t^{z_1 - 1}}{(1 + t)^{z_1 + z_2}} \ dt }$$
$${\frac{1}{a} B\left(b-\frac{s}{a},\frac{s}{a}\right) = \int_{0}^{\infty} \frac{t^{s - 1}}{(1 + t^a)^{b}}\ dt }$$
Reciprocal Beta Function:
$${\int_{0}^{\frac{\pi}{2}} (\cos t)^{x - 1} \cos yt \ dt = \frac{\pi}{2^x x B \left( \frac{x+y+1}{2} , \frac{x-y+1}{2} \right) } }$$
$${\int_{0}^{\pi} (\sin t)^{x - 1} \sin yt \ dt = \frac{\pi \sin \left( \frac{\pi y}{2} \right) }{2^{x-1} x B \left( \frac{x+y+1}{2} , \frac{x-y+1}{2} \right) } }$$
$${\int_{0}^{\pi} (\sin t)^{x - 1} \cos yt \ dt = \frac{\pi \cos \left( \frac{\pi y}{2} \right) }{2^{x-1} x B \left( \frac{x+y+1}{2} , \frac{x-y+1}{2} \right) } }$$
$${\int_{0}^{\pi} (\cos t)^{x - 1} \sin yt \ dt = \frac{\pi \cos \left( \frac{\pi y}{2} \right) }{2^{x-1} x B \left( \frac{x+y+1}{2} , \frac{x-y+1}{2} \right) } }$$
$${\int \sin^n(x) \ dx = \frac{n - 1}{n} \int \sin^{n-2}(x) \ dx - \frac{\sin^{n-1}(x) \ \cos(x)}{n}}$$
$${\int \cos^n(x) \ dx = \frac{\cos^{n-1}(x) \ \sin(x)}{n} + \frac{n - 1}{n} \int \cos^{n-2}(x) \ dx}$$
$${n \in \mathbb{Z}^+}$$
$${k, m \in \mathbb{Z}}$$
$${a = \frac{k \pi}{2}}$$
$${b = \frac{m \pi}{2}}$$
When n is odd:
$${I_n = \int_{a}^{b} \sin^n(x) \ dx = \int_{a}^{b} \cos^n(x) \ dx = \frac{(n - 1)!!}{n!!} I_1}$$
When n is even:
$${I_n = \int_{a}^{b} \sin^n(x) \ dx = \int_{a}^{b} \cos^n(x) \ dx = \frac{(n - 1)!!}{n!!} I_0}$$
$${m,n \in \mathbb{Z}^+}$$
When m and n are even:
$${\int_{0}^{\frac{\pi}{2}} \sin^n(x) \cos^m(x) \ dx = \frac{(n - 1)!! (m - 1)!!}{(m+n)!!} \frac{\pi}{2}}$$
$${\int_{0}^{\frac{\pi}{2}} \sin^n(x) \ dx = \frac{(n - 1)!!}{n!!} \frac{\pi}{2}}$$
$${\int_{0}^{\frac{\pi}{2}} \cos^m(x) \ dx = \frac{(m - 1)!!}{m!!} \frac{\pi}{2}}$$
When either m or n is odd:
$${\int_{0}^{\frac{\pi}{2}} \sin^n(x) \cos^m(x) \ dx = \frac{(n - 1)!! (m - 1)!!}{(m+n)!!} }$$
$${\int_{0}^{\frac{\pi}{2}} \sin^n(x) \ dx = \frac{(n - 1)!!}{n!!}}$$
$${\int_{0}^{\frac{\pi}{2}} \cos^m(x) \ dx = \frac{(m - 1)!!}{m!!}}$$
$${\int_{a}^{b} f(x)\ dx = \int_{a}^{b} f(b + a - x) \ dx}$$
$${\text{if } f(2a-x) = f(x) }$$
$${\int_{0}^{2a} f(x) \ dx = 2 \int_{0}^{a} f(x) \ dx}$$
$${\text{if } f(2a-x) = -f(x) }$$
$${\int_{0}^{2a} f(x) \ dx = 0}$$
$${\text{if } f(x+a) = f(x) }$$
$${\int_{0}^{na} f(x) \ dx = n \int_{0}^{a} f(x) \ dx}$$
$${\text{if } f(x+a) = -f(x) }$$
$${\int_{0}^{2a} f(x) \ dx = 0}$$
$${f(x) \text{ is an even function.}}$$
$${\int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) \ dx}$$
$${f(x) \text{ is an odd function.}}$$
$${\int_{-a}^{a} f(x) \ dx = 0}$$
$${\text{if } f(\pi \pm x) = f(x) }$$
$${\int_{0}^{\infty} \frac{\sin^2(x)}{x^2} f(x) \ dx = \int_{0}^{\infty} \frac{\sin(x)}{x} f(x) \ dx = \int_{0}^{\frac{\pi}{2}} f(x) \ dx}$$
$${PV \int_{-\infty}^{\infty} F(u) \ dx = PV \int_{-\infty}^{\infty} F(x) \ dx}$$
$${u = x - a - \sum_{k = 1}^{n} \frac{|a_k|}{x - b_k} }$$
Cauchy-Schlomilch:
$${u = x - \frac{a}{x}, x \gt 0 }$$
$${\int_{0}^{\infty} \frac{f(ax) - f(bx)}{x} \ dx = \left( f(\infty) - f(0) \right) \ln \left( \frac{a}{b} \right)}$$
$${\int_{-\infty}^{\infty} f(x) \delta(x - a) \ dx = f(a)}$$