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Integration Formulas

Indefinite Integration Formulas

\( \int k \, dx = kx + C \)

Example: \( \int 5 \, dx = 5x + C \)

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

Example: \( \int x^2 \, dx = \frac{{x^3}}{3} + C \)

\( \int e^x \, dx = e^x + C \)

Example: \( \int e^x \, dx = e^x + C \)

\( \int \sin(x) \, dx = -\cos(x) + C \)

Example: \( \int \sin(x) \, dx = -\cos(x) + C \)

\( \int \cos(x) \, dx = \sin(x) + C \)

Example: \( \int \cos(x) \, dx = \sin(x) + C \)

\( \int \tan(x) \, dx = -\ln|\cos(x)| + C \)

Example: \( \int \tan(x) \, dx = -\ln|\cos(x)| + C \)

\( \int \cot(x) \, dx = \ln|\sin(x)| + C \)

Example: \( \int \cot(x) \, dx = \ln|\sin(x)| + C \)

\( \int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C \)

Example: \( \int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C \)

\( \int \csc(x) \, dx = -\ln|\csc(x) + \cot(x)| + C \)

Example: \( \int \csc(x) \, dx = -\ln|\csc(x) + \cot(x)| + C \)

\( \int \sec^2(x) \, dx = \tan(x) + C \)

Example: \( \int \sec^2(x) \, dx = \tan(x) + C \)

\( \int \csc^2(x) \, dx = -\cot(x) + C \)

Example: \( \int \csc^2(x) \, dx = -\cot(x) + C \)

\( \int \sec(x) \tan(x) \, dx = \sec(x) + C \)

Example: \( \int \sec(x) \tan(x) \, dx = \sec(x) + C \)

\( \int \csc(x) \cot(x) \, dx = -\csc(x) + C \)

Example: \( \int \csc(x) \cot(x) \, dx = -\csc(x) + C \)

Example: \( \int \frac{dx}{1+x^2} = \tan^{-1}(x) + C \)

Example: \( \int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1}(x) + C \)

Example: \( \int \frac{dx}{x\sqrt{x^2 - 1}} = \sec^{-1}(x) + C \)

Example: \( \int \frac{dx}{x^2-1} = \frac{1}{2} \ln \left|\frac{x-1}{x+1}\right| + C \)

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

Example: \( \int \frac{1}{x} \, dx = \ln|x| + C \)

\( \int e^{ax} \, dx = \frac{e^{ax}}{a} + C \)
\( \int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \)
\( \int \frac{x}{a^2 + x^2} \, dx = \frac{1}{2} \ln(a^2 + x^2) + C \)
\( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right) + C \)
\( \int \frac{1}{x\sqrt{x^2 - a^2}} \, dx = \frac{1}{a} \sec^{-1}\left(\frac{x}{a}\right) + C \)
\( \int \frac{x}{\sqrt{x^2 + a^2}} \, dx = \sqrt{x^2 + a^2} + C \)
\( \int \frac{1}{(x-a)(x-b)} \, dx = \frac{1}{b-a} \ln\left|\frac{x-a}{x-b}\right| + C \)
\( \int \frac{1}{(x^2 - a^2)} \, dx = \frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right| + C \)
\( \int \frac{x^2}{(x^2 + a^2)^2} \, dx = \frac{1}{2a^2} \left( \frac{x}{x^2 + a^2} + \tan^{-1}\left(\frac{x}{a}\right)\right) + C \)
\( \int x^n \ln(x) \, dx = \frac{x^{n+1}}{n+1} \ln(x) - \frac{x^{n+1}}{(n+1)^2} + C \)
\( \int \ln(x) \, dx = x \ln(x) - x + C \)
\( \int \frac{\ln(x)}{x} \, dx = \frac{(\ln(x))^2}{2} + C \)
\( \int \ln(x^2 + 1) \, dx = x \ln(x^2 + 1) - 2x + 2\tan^{-1}(x) + C \)
\( \int \frac{1}{x \ln(x)} \, dx = \ln(\ln(x)) + C \)
\( \int \frac{\ln(x)}{x^2} \, dx = -\frac{\ln(x)}{x} - \frac{1}{x} + C \)