Developed by Alex theMighty for EDUCATION PURPOSES
| Rule | Basic Form (dx) | General Form (u) |
|---|---|---|
| A. Constant Multiple | $\int k \cdot f(x) \, dx = k \int f(x) \, dx$ | |
| B. Sum / Difference | $\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$ | |
| C. Constant Rule | $\int a \, dx = ax + C$ | $\int a \, du = au + C$ |
| 1. Power Rule | $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$ | $\int u^n \, dx = \frac{1}{u'} \frac{u^{n+1}}{n+1} + C \quad (n \neq -1)$ |
| 2. 1/u Rule | $\int \frac{1}{x} \, dx = \ln|x| + C$ | $\int \frac{1}{u} \, dx = \frac{1}{u'} \ln|u| + C$ |
| 3. Exponential (Base $e$) | $\int e^x \, dx = e^x + C$ | $\int e^u \, dx = \frac{1}{u'} e^u + C$ |
| 4. Exponential (Base $a$) | $\int a^x \, dx = \frac{a^x}{\ln a} + C$ | $\int a^u \, dx = \frac{1}{u'} \frac{a^u}{\ln a} + C$ |
| 5. Sine | $\int \sin x \, dx = -\cos x + C$ | $\int \sin u \, dx = -\frac{1}{u'} \cos u + C$ |
| 6. Cosine | $\int \cos x \, dx = \sin x + C$ | $\int \cos u \, dx = \frac{1}{u'} \sin u + C$ |
| 7. Inverse Cosine Squared | $\int \frac{1}{\cos^2 x} \, dx = \tan x + C$ | $\int \frac{1}{\cos^2 u} \, dx = \frac{1}{u'} \tan u + C$ |
| 8. Inverse Sine Squared | $\int \frac{1}{\sin^2 x} \, dx = -\cot x + C$ | $\int \frac{1}{\sin^2 u} \, dx = -\frac{1}{u'} \cot u + C$ |
| 9. Tangent | $\int \tan x \, dx = -\ln|\cos x| + C$ | $\int \tan u \, dx = -\frac{1}{u'} \ln|\cos u| + C$ |
| 10. Cotangent | $\int \cot x \, dx = \ln|\sin x| + C$ | $\int \cot u \, dx = \frac{1}{u'} \ln|\sin u| + C$ |