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Integrals Cheat Sheet

 

Common Integrals

x1dx=ln(x) 1x dx=ln(x)
|x|dx=xx22  exdx=ex
sin(x)dx=cos(x) cos(x)dx=sin(x)
xadx=xa+1a+1 ,      a1


Trigonometric Integrals

sec2(x)dx=tan(x) csc2(x)dx=cot(x)
1sin2(x) dx=cot(x) 1cos2(x) dx=tan(x)


Arc Trigonometric Integrals

1x2+1 dx=arctan(x) 1x2+1 dx=arccot(x)
11x2 dx=arcsin(x) 11x2 dx=arccos(x)
1|x|x21 dx=arcsec(x) 1|x|x21 dx=arccsc(x)
1x2+1 dx=arcsinh(x) 11x2 dx=arctanh(x)
1|x|x2+1 dx=arccsch(x)


Hyperbolic Integrals

sech2(x)dx=tanh(x) csch2(x)dx=(coth(x))
cosh(x)dx=sinh(x) sinh(x)dx=cosh(x)
csch(x)dx=ln(tanh(x2 )) sec(x)dx=ln(tan(x)+sec(x))


Integrals of Special Functions

cos(x2π2 )dx=(x) sin(x)x dx=Si(x)
cos(x)x dx=Ci(x) sinh(x)x dx=Shi(x)
cosh(x)x dx=Chi(x) exp(x)x dx=Ei(x)
expx2dx=π2 erf(x) expx2dx=expx2F(x)
sin(x2π2 )dx=S(x) sin(x2)dx=π2 S(2π x)
1ln(x) dx=li(x)


Indefinite Integrals Rules

Integration By Parts uv=uv uv
Integral of a constant f(a)dx=x·f(a)
Take the constant out a·f(x)dx=a·f(x)dx
Sum Rule f(x)±g(x)dx=f(x)dx±g(x)dx
Add a constant to the solution
If dF(x)dx =f(x) then f(x)dx=F(x)+C
Power Rule xadx=xa+1a+1 ,      a1
Integral Substitution f(g(x))·g(x)dx=f(u)du,     u=g(x)


Definite Integrals Rules

Definite Integral Boundaries
abf(x)dx=F(b)F(a)
=limxb(F(x))limxa+(F(x))
Odd function If f(x)=f(x)aaf(x)dx=0
Undefined points
If exist b, a<b<c, f(b)=undefined,
ac f(x)dx=ab f(x)dx+bc f(x)dx
Same points defined aa f(x)dx=0