\int f'(x) dx = f(x) + const
| function | integration with respect to x |
|---|---|
| a | ax + const |
| x^n | \frac{1}{n+1} x^{n+1} + const |
| e^x | e^x + const |
| a^x | \frac{a^x}{\ln(a)} + const |
| \ln(x) | x\ln(x) - x + const |
| \sin(x) | -\cos(x) + const |
| \cos(x) | \sin(x) + const |
| \int af(x) | a \int f(x) |
Let u = g(x) then: \int f(g(x))g'(x) dx = \int f(u)du.
Example:
Solve \int cos(x^2)x. Let u = x^2, u' = 2x.
Then \int cos(x^2)x = \frac{1}{2} \int cos(x^2)2x = \frac{1}{2} sin(x^2)