Handy Formulas
`int a dx = a x`
`int x^a dx = x^(a+1)/(a+1)`, provided `a ne -1`
`int e^x dx = e^x`
`int 1/x dx = ln|x|`
`int sqrt(a^2-x^2) dx = x/2sqrt(a^2-x^2)+a^2/2sin^-1(x/a)`
For `a < b < c`, we have `int_a^c g(x) dx = int_a^b g(x) dx + int_b^c g(x) dx.`
Integration by Parts formula: `int u dv = uv - int v du.`
Please contribute more formulas!
Trick For Calculating Integrals
Suppose you have a complicated integral, but you can express it as
Complicated integral `= int c f(x) dx `
where `c` is some constant and `f(x)` is a probability density function for some random variable. Then we know that `int f(x) dx = 1`, so
Complicated integral `=c`
For example, suppose we have
`int_{x>0} 3 exp(-x/2) dx `
Write this as
`int_{x>0} 6 * (1/2) exp(-x/2) dx = 6 * int_{x>0} (1/2) exp(-x/2) dx `
The integral on the right is an integral of an exponential density (with `beta=2`). So the answer is 6.
In this case, you could just as easily calculate the integral directly! But the trick comes in handy from time to time.