Calculus: Integrals & Antiderivatives

F is an antiderivative of f if F' = f . Constants differentiate to 0, so every antiderivative is only determined up to a constant C .

\dfrac{x^{n+1}}{n + 1} + C

\ln|x| + C

e^x + C

\dfrac{a^x}{\ln a} + C

\sin x + C

-\cos x + C

\tan x + C

\sec x + C

-\ln|\cos x| + C = \ln|\sec x| + C

\arctan x + C

\dfrac{1}{a}\arctan\dfrac{x}{a} + C

\arcsin x + C

\dfrac{d}{dx} \displaystyle\int_a^x f(t)\,dt = f(x)

\displaystyle\int_a^b f(x)\,dx = F(b) - F(a) , where F' = f

f\big(g(x)\big) \cdot g'(x)

\displaystyle\int f\big(g(x)\big)\,g'(x)\,dx = \int f(u)\,du with u = g(x) , du = g'(x)\,dx

\displaystyle\int u\,dv = uv - \int v\,du

\displaystyle\int_a^a f\,dx = 0 \displaystyle\int_b^a f\,dx = -\int_a^b f\,dx \displaystyle\int_a^b f\,dx + \int_b^c f\,dx = \int_a^c f\,dx

8

1

2

\sin\left(x^2\right) + C

x e^x - e^x + C

x\ln x - x + C

f_{\text{avg}} = \dfrac{1}{b - a}\displaystyle\int_a^b f(x)\,dx

\displaystyle\int_a^b \big[\text{top}(x) - \text{bottom}(x)\big]\,dx

Disk: V = \pi\displaystyle\int_a^b R(x)^2\,dx Washer: V = \pi\displaystyle\int_a^b \left[R(x)^2 - r(x)^2\right]dx

Displacement: \displaystyle\int_a^b v(t)\,dt Total distance: \displaystyle\int_a^b |v(t)|\,dt

\dfrac{\Delta x}{2}\big[f(x_0) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(x_n)\big]