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Math formulas: Common integrals

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Indefinite Integral

Method of substitution

f(g(x))g(x)dx=f(u)du \int f\left(g(x)\right)\cdot g'(x) dx = \int f(u) du

Integration by parts

f(x)g(x)dx=f(x)g(x)g(x)f(x)dx \int f(x) \cdot g'(x)dx = f(x) \cdot g(x) - \int g(x) \cdot f'(x)dx

Integrals of Rational and Irrational Functions

xndx=xn+1n+1+C,n1 \int x^n dx = \frac{x^{n+1}}{n+1} + C , n \ne 1
1xdx=lnx+C \int \frac{1}{x} dx = \ln|x| + C
cdx=cx+C \int c \, dx = c \cdot x + C
xdx=x22+C \int x \, dx = \frac{x^2}{2} + C
x2dx=x33+C \int x^2 \, dx = \frac{x^3}{3} + C
1x2dx=1x+C \int \frac{1}{x^2} dx = -\frac{1}{x} + C
xdx=2xx3+C \int \sqrt{x} \, dx = \frac{2\cdot x \cdot \sqrt{x} }{3} + C
11+x2dx=arctanx+C \int \frac{1}{1+x^2} dx = \arctan x + C
11x2dx=arcsinx+C \int \frac{1}{\sqrt{1-x^2}} dx = \arcsin x + C

Integrals of Trigonometric Functions

sinxdx=cosx+C \int \sin x\,dx = -\cos x + C
cosxdx=sinx+C \int \cos x\,dx = \sin x + C
tanxdx=lnsecx+C \int \tan x\,dx = \ln|\sec x| + C
secxdx=lntanx+secx+C \int \sec x\,dx = \ln|\tan x + \sec x | + C
sin2xdx=12(xsinxcosx)+C \int \sin^2x\,dx = \frac{1}{2}(x-\sin x \cdot \cos x) + C
cos2xdx=12(x+sinxcosx)+C \int \cos^2x\,dx = \frac{1}{2}(x + \sin x \cdot \cos x) + C
tan2xdx=tanxx+C \int \tan^2x\,dx = \tan x - x + C
sec2xdx=tanx+C \int \sec^2x\,dx = \tan x + C

Integrals of Exponential and Logarithmic Functions

lnxdx=xlnxx+C \int \ln x \,dx =x \cdot \ln x -x + C
xnlnxdx=xn+1n+1lnxxn+1(n+1)2+C \int x^n \cdot \ln x \,dx =\frac{x^{n+1}}{n+1} \ln x - \frac{x^{n+1}}{(n+1)^2} + C
exdx=ex+C \int e^x\,dx = e^x + C
axdx=axlna+C \int a^x\,dx = \frac{a^x}{\ln a} + C

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