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Integrals of Common Functions

The antiderivatives you’ll meet most often. Open any row for the technique, the steps and a graph of the +C family.

Integral of ln x $\int \ln\left(x\right)\,dx = x \ln\left(x\right) - x + C$
Integral of 1/x $\int \frac{1}{x}\,dx = \ln\left(\left|x\right|\right) + C$
Integral of sin x $\int \sin\left(x\right)\,dx = -\cos\left(x\right) + C$
Integral of cos x $\int \cos\left(x\right)\,dx = \sin\left(x\right) + C$
Integral of tan x $\int \tan\left(x\right)\,dx = -\ln\left(\left|\cos\left(x\right)\right|\right) + C$
Integral of eˣ $\int e^{x}\,dx = e^{x} + C$
Integral of sec x $\int \sec\left(x\right)\,dx = \ln\left(\left|\sec\left(x\right) + \tan\left(x\right)\right|\right) + C$
Integral of sec²x $\int \sec^{2}\left(x\right)\,dx = \tan\left(x\right) + C$
Integral of x·eˣ $\int x e^{x}\,dx = x e^{x} - e^{x} + C$
Integral of sin²x $\int \sin^{2}\left(x\right)\,dx = \frac{1}{2}x - \frac{1}{4}\sin\left(2 x\right) + C$
Integral of cos²x $\int \cos^{2}\left(x\right)\,dx = \frac{1}{2}x + \frac{1}{4}\sin\left(2 x\right) + C$
Integral of √x $\int \sqrt{x}\,dx = \frac{2}{3}x^{\frac{3}{2}} + C$
Integral of 1/(1 + x²) $\int \frac{1}{x^{2} + 1}\,dx = \arctan\left(x\right) + C$
Integral of e⁻ˣ $\int e^{-x}\,dx = -e^{-x} + C$
Integral of x ln x $\int x \ln\left(x\right)\,dx = \frac{1}{2}x^{2} \ln\left(x\right) - \frac{1}{4}x^{2} + C$

Need a function that isn’t listed? Use the integral calculator, or see the rules behind them on integration rules.