Integrals and Integral Shortcuts

An example of an integral.  I will be using "I" instead of the integral sign at the beginning.

If the integral of a function f(x) is f'(x), the integral of the function is f(x).  Thus, the integral is the "antiderivative" of f'(x).

The shortcuts for integrals are usually more helpful than those for derivatives, since integrals are more complex to calculate by hand.  Some shortcuts include:

I(u^n du) = u^(n+1)/(n+1) when n != -1
I(du/u) = ln(u)
I(sin(u) du) = -cos(u)
I(cos(u) du) = sin(u)
I(tan(u) du) = ln|sec(u)|
I(cot(u) du) = ln|sin(u)|
I(sec(u) du) = ln|sec(u) + tan(u)|
I(csc(u) du) = ln|csc(u) - cot(u)|
I(sec(u)^2 du) = tan(u)
I(csc(u)^2 du) = -cot(u)
I(sec(u)tan(u) du) = sec(u)
I(csc(u)cot(u) du) = -csc(u)
I(a^u du) = a^u/ln(a)
I(e^u du) = e^u
I(sinh(u) du) = cosh(u)
I(cosh(u) du) = sinh(u)
I(tanh(u) du) = ln(cosh(u))
I(coth(u) du) = ln|sinh(u)|
I(sech(u) du) = arctan(sinh(u))
I(csch(u) du) = -arccoth(cosh(u))
I(sech(u)^2 du) = tanh(u)
I(csch(u)^2 du) = -coth(u)
I(sech(u)tanh(u) du) = -csch(u)
I(du/(a^2-u^2)^.5 = arcsin(u/a)
I(du/(u^2+-a^2)^.5) = ln|u + (u^2 +- a^2)^.5|
I(du/(u^2+a^2)) = 1/a arctan(u/a)
I(du/(u^2-a^2)) = 1/(2a) ln|(u-a)/(u+a)|
I(du/(u(a^2+-u^2)^.5)) = 1/a(ln|u/(a+(a^2+-u^2)^.5)|
I(du/(u(u^2-a^2)) = 1/a arccos(a/u)
I((u^2+-a^2)^.5 du) = u/2(u^2+-a^2)^.5 +- a^2/2 ln|u + (u^2+-a^2)^.5|
I((a^2-u^2)^.5 du) = u/2(a^2-u^2)^.5 + a^2/2 arcsin(u/a)
I(e^(au) sin(bu) du) = e^(au)(a sin(bu) - b cos(bu))/(a^2+b^2)