Integrals Cheat Sheet
∫x−1dx=ln(x)
∫1x dx=ln(x)
∫|x|dx=x√x22
∫exdx=ex
∫sin(x)dx=−cos(x)
∫cos(x)dx=sin(x)
∫xadx=xa+1a+1 , a≠−1
∫sec2(x)dx=tan(x)
∫csc2(x)dx=−cot(x)
∫1sin2(x) dx=−cot(x)
∫1cos2(x) dx=tan(x)
∫1x2+1 dx=arctan(x)
∫−1x2+1 dx=arccot(x)
∫1√1−x2 dx=arcsin(x)
∫−1√1−x2 dx=arccos(x)
∫1|x|√x2−1 dx=arcsec(x)
∫−1|x|√x2−1 dx=arccsc(x)
∫1√x2+1 dx=arcsinh(x)
∫11−x2 dx=arctanh(x)
∫1|x|√x2+1 dx=−arccsch(x)
∫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))
∫cos(x2π2 )dx=ℂ(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)
∫exp−x2dx=√π2 erf(x)
∫expx2dx=expx2F(x)
∫sin(x2π2 )dx=S(x)
∫sin(x2)dx=√π2 S(√2π x)
∫1ln(x) dx=li(x)
Integration By Parts
∫ uv′=uv−∫ u′v
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 , a≠−1
Integral Substitution
∫f(g(x))·g′(x)dx=∫f(u)du, u=g(x)
Definite Integral Boundaries
∫abf(x)dx=F(b)−F(a)
=limx→b−(F(x))−limx→a+(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