Antiderivatives and Indefinite Integrals

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$)

$\int \frac{1}{x}\,dx = \ln|x| + C$

$\int e^x\,dx = e^x + C$, $\int a^x\,dx = \frac{a^x}{\ln a} + C$

$\int \sin x\,dx = -\cos x + C$, $\int \cos x\,dx = \sin x + C$

$\int \sec^2 x\,dx = \tan x + C$, $\int \csc^2 x\,dx = -\cot x + C$

$\int \frac{1}{1+x^2}\,dx = \arctan x + C$, $\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C$

Given $f'(x)$ and $f(a) = b$: integrate, then use the condition to find $C$.

$f'(x) = 3x^2 - 2$, $f(1) = 4$.

$f(x) = x^3 - 2x + C$. $f(1) = 1 - 2 + C = 4$ → $C = 5$.

$f(x) = x^3 - 2x + 5$.