MAC 2312 Common Integration Formulas

Vocabulary flashcards covering the common integration formulas for MAC 2312.

Integration of Composite Functions Formula

$$\int f'(g(x)) g'(x) \, dx = f(g(x)) + C$$

Integration by Parts Formula

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

Power Rule for Integration

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

Integral of $\frac{1}{x}$

$$\ln|x| + C$$

Integral of $e^x$

$$e^x + C$$

Integral of $a^x$

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

Integral of $\ln(x)$

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

Integral of $\sin(x)$

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

Integral of $\cos(x)$

$$\sin(x) + C$$

Integral of $\sec^2(x)$

$$\tan(x) + C$$

Integral of $\csc^2(x)$

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

Integral of $\sec(x) \tan(x)$

$$\sec(x) + C$$

Integral of $\csc(x) \cot(x)$

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

Integral of $\tan(x)$

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

Integral of $\cot(x)$

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

Integral of $\sec(x)$

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

Integral of $\csc(x)$

$$-\ln|\csc(x) + \cot(x)| + C = \ln|\csc(x) - \cot(x)| + C$$

Integral of $\sec^3(x)$

$$\frac{1}{2} (\sec(x) \tan(x) + \ln|\sec(x) + \tan(x)|) + C$$

Integral of $\csc^3(x)$

$$-\frac{1}{2} (\csc(x) \cot(x) + \ln|\csc(x) + \cot(x)|) + C$$

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

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

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

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

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

$$\frac{1}{a} \sec^{-1}\left(\frac{x}{a}\right) + C$$