Integrals Cheat Sheet


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Common Integrals

\int x^{-1}dx=\ln(x) \int \frac{1}{x} dx=\ln(x)
\int |x|dx=\frac{x\sqrt{{x}^2}}{2} \int e^{x}dx=e^{x}
\int \sin(x)dx=-\cos(x) \int \cos(x)dx=\sin(x)
\int x^{a}dx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1


Trigonometric Integrals

\int \sec^2(x) dx=\tan(x) \int \csc^2(x) dx =-\cot(x)
\int \frac{1}{\sin^2(x)}dx=-\cot(x) \int \frac{1}{\cos^2(x)}dx=\tan(x)


Arc Trigonometric Integrals

\int \frac{1}{{x}^2+1}dx=\arctan(x) \int \frac{-1}{{x}^2+1}dx=\arccot(x)
\int \frac{1}{\sqrt{1-{x}^2}}dx=\arcsin(x) \int \frac{-1}{\sqrt{1-{x}^2}}dx=\arccos(x)
\int \frac{1}{|x|\sqrt{{x}^2-1}} dx = \arcsec(x) \int \frac{-1}{|x|\sqrt{{x}^2-1}} dx = \arccsc(x)
\int \frac{1}{\sqrt{{x}^2+1}} dx = \arcsinh(x) \int \frac{1}{1-{x}^2} dx = \arctanh(x)
\int \frac{1}{|x|\sqrt{{x}^2+1}} dx = -\arccsch(x)


Hyperbolic Integrals

\int \sech^2(x) dx = \tanh(x) \int \csch^2(x) dx = (-\coth(x))
\int \cosh(x) dx = \sinh(x) \int \sinh(x) dx = \cosh(x)
\int \csch(x) dx = \ln(\tanh(\frac{x}{2})) \int \sec(x) dx = \ln(\tan(x)+\sec(x))


Integrals of Special Functions

\int \cos(\frac{{x}^2\pi}{2})dx = \C(x) \int \frac{\sin (x)}{x}dx = \Si(x)
\int \frac{\cos (x)}{x}dx = \Ci(x) \int \frac{\sinh (x)}{x}dx = \Shi(x)
\int \frac{\cosh (x)}{x}dx = \Chi(x) \int \frac{\exp (x)}{x}dx = \Ei(x)
\int \exp{-{x}^2}dx = \frac{\sqrt{\pi}}{2}\erf(x) \int \exp{{x}^2}dx = \exp{{x}^2}\F(x)
\int \sin(\frac{{x}^2\pi}{2})dx = \S(x) \int \sin({x}^2)dx = \sqrt{\frac{\pi}{2}}\S(\sqrt{\frac{2}{\pi}}x)
\int \frac{1}{\ln(x)}dx=\li(x)


Indefinite Integrals Rules

Integration By Parts \int \:uv'=uv-\int \:u'v
Integral of a constant \int f\left(a\right)dx=x\cdot f\left(a\right)
Take the constant out \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx
Sum Rule \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx
Add a constant to the solution \mathrm{If} \frac{dF(x)}{dx}=f(x) \mathrm{then} \int{f(x)}dx=F(x)+C
Power Rule \int x^{a}dx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1
Integral Substitution \int f\left(g\left(x\right)\right)\cdot g^'\left(x\right)dx=\int f\left(u\right)du,\:\quad u=g\left(x\right)


Definite Integrals Rules

Definite Integral Boundaries
\int_{a}^{b}f(x)dx=F(b)-F(a)
=\lim_{x\to b-}(F(x))-\lim _{x\to a+}(F(x))
Odd function \mathrm{If }f\left(x\right)=-f\left(-x\right)\Rightarrow\int _{-a}^{a}f(x)dx=0
Undefined points
\mathrm{If\:exist\:b,\:a\lt b\lt c,\:and}\:f\left(b\right)=\mathrm{undefined},
\mathrm{Then}\:\int _a^c\:f\left(x\right)dx=\int _a^b\:f\left(x\right)dx+\int _b^c\:f\left(x\right)dx
Same points defined \int _a^a\:f\left(x\right)dx=0


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