Standard Integration Formulas

A complete review set of standard indefinite integration rules and inverse trigonometric form integrals.

Integral of Constant Function $\int a\,dx$

$$ax + c$$

Power Rule for Integration $\int x^n\,dx$

$\frac{x^{n+1}}{n+1} + c$ where $(n \neq -1)$

Integral of $\int \frac{1}{x}\,dx$

$$\ln|x| + c$$

Integral of Exponential $\int e^x\,dx$

$$e^x + c$$

Integral of General Exponential $\int a^x\,dx$

$$\frac{a^x}{\ln(a)} + c$$

Integral of Natural Logarithm $\int \ln(x)\,dx$

$$x\ln(x) - x + c$$

Integral of Sine $\int \sin(x)\,dx$

$$-\cos(x) + c$$

Integral of Cosine $\int \cos(x)\,dx$

$$\sin(x) + c$$

Integral of Tangent $\int \tan(x)\,dx$

$$\ln|\sec(x)| + c$$

Integral of Cotangent $\int \cot(x)\,dx$

$$\ln|\sin(x)| + c$$

Integral of Secant $\int \sec(x)\,dx$

$$\ln|\sec(x) + \tan(x)| + c$$

Integral of Cosecant $\int \csc(x)\,dx$

$$-\ln|\csc(x) + \cot(x)| + c$$

Integral of Secant Squared $\int \sec^2(x)\,dx$

$$\tan(x) + c$$

Integral of Secant Tangent $\int \sec(x)\tan(x)\,dx$

$$\sec(x) + c$$

Integral of Cosecant Squared $\int \csc^2(x)\,dx$

$$-\cot(x) + c$$

Integral of Cosecant Cotangent $\int \csc(x)\cot(x)\,dx$

$$-\csc(x) + c$$

Integral of Tangent Squared $\int \tan^2(x)\,dx$

$$\tan(x) - x + c$$

Integral of $\int \frac{1}{a^2 + x^2}\,dx$

$$\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + c$$

Integral of $\int \frac{1}{\sqrt{a^2 - x^2}}\,dx$

$$\sin^{-1}\left(\frac{x}{a}\right) + c$$

Integral of $\int \frac{1}{x\sqrt{x^2 - a^2}}\,dx$

$$\frac{1}{a}\sec^{-1}\left|\frac{x}{a}\right| + c$$