Indefinite Integration Formulas
1. Constant Rule
\( \int k \, dx = kx + C \)
Example: \( \int 5 \, dx = 5x + C \)
2. Power Rule
\( \int x^n \, dx = \frac{{x^{n+1}}}{{n+1}} + C \), for \( n \neq -1 \)
Example: \( \int x^2 \, dx = \frac{{x^3}}{3} + C \)
3. Exponential Rule
\( \int e^x \, dx = e^x + C \)
Example: \( \int e^x \, dx = e^x + C \)
4. Trigonometric Functions
\( \int \sin(x) \, dx = -\cos(x) + C \)
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: \( \int \csc^2(x) \, dx = -\cot(x) + C \)
- \( \int \sec(x) \tan(x) \, dx = \sec(x) + C \)
- \( \int \csc(x) \cot(x) \, dx = -\csc(x) + C \)
Example: \( \int \sec(x) \tan(x) \, dx = \sec(x) + C \)
Example: \( \int \csc(x) \cot(x) \, dx = -\csc(x) + C \)
5. Inverse Trigonometric Functions
- \( \int \frac{dx}{1+x^2} = \tan^{-1}(x) + C \)
- \( \int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1}(x) + C \)
- \( \int \frac{dx}{x\sqrt{x^2 - 1}} = \sec^{-1}(x) + C \)
- \( \int \frac{dx}{x^2-1} = \frac{1}{2} \ln \left|\frac{x-1}{x+1}\right| + C \)
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: \( \int \frac{dx}{x^2-1} = \frac{1}{2} \ln \left|\frac{x-1}{x+1}\right| + C \)
6. Logarithmic Functions
\( \int \frac{1}{x} \, dx = \ln|x| + C \)
Example: \( \int \frac{1}{x} \, dx = \ln|x| + C \)
7. Special Cases
- \( \int e^{ax} \, dx = \frac{e^{ax}}{a} + C \)
- \( \int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \)
- \( \int \frac{x}{a^2 + x^2} \, dx = \frac{1}{2} \ln(a^2 + x^2) + C \)
- \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right) + C \)
- \( \int \frac{1}{x\sqrt{x^2 - a^2}} \, dx = \frac{1}{a} \sec^{-1}\left(\frac{x}{a}\right) + C \)
- \( \int \frac{x}{\sqrt{x^2 + a^2}} \, dx = \sqrt{x^2 + a^2} + C \)
- \( \int \frac{1}{(x-a)(x-b)} \, dx = \frac{1}{b-a} \ln\left|\frac{x-a}{x-b}\right| + C \)