SJMathTube Formula Sheet

Integration Formulae

Integration means continuous summation or accumulation. A definite integral gives signed accumulation, not always ordinary area.

Concept Note

\[ \int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^{n} f(x_i^*)\Delta x \]

For example, \[ \int_0^{2\pi}\sin x\,dx=0 \] but the total geometrical area is \[ \int_0^{2\pi}|\sin x|\,dx=4 \]

Important Note

\[ \text{If } \int f(x)\,dx=\phi(x)+C, \] \[ \text{then } \int f(Ax)\,dx=\frac{\phi(Ax)}{A}+C \]

1. Basic Algebraic Formulae

\[ \int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\quad n\ne -1 \]

\[ \int \frac{1}{x}\,dx=\log|x|+C \]

\[ \int e^x\,dx=e^x+C \]

\[ \int a^x\,dx=\frac{a^x}{\log a}+C \]

\[ \int \frac{1}{\sqrt{x}}\,dx=2\sqrt{x}+C \]

\[ \int \frac{1}{x^2}\,dx=-\frac{1}{x}+C \]

\[ \int dx=x+C \]

2. Trigonometric Formulae

\[ \int \sin x\,dx=-\cos x+C \]

\[ \int \cos x\,dx=\sin x+C \]

\[ \int \sec^2 x\,dx=\tan x+C \]

\[ \int \csc^2 x\,dx=-\cot x+C \]

\[ \int \sec x\tan x\,dx=\sec x+C \]

\[ \int \csc x\cot x\,dx=-\csc x+C \]

3. Logarithmic Trigonometric Formulae

\[ \int \tan x\,dx=\log|\sec x|+C \]

\[ \int \cot x\,dx=\log|\sin x|+C \]

\[ \int \sec x\,dx=\log|\sec x+\tan x|+C \]

\[ \int \csc x\,dx=\log|\csc x-\cot x|+C \]

4. Inverse Trigonometric Forms

\[ \int \frac{1}{x^2+1}\,dx=\tan^{-1}x+C \]

\[ \int \frac{1}{\sqrt{1-x^2}}\,dx=\sin^{-1}x+C \]

5. Special Integrals

\[ \int \frac{1}{x^2+A^2}\,dx = \frac{1}{A}\tan^{-1}\left(\frac{x}{A}\right)+C \]

\[ \int \frac{1}{x^2-A^2}\,dx = \frac{1}{2A}\log\left|\frac{x-A}{x+A}\right|+C \]

\[ \int \frac{1}{A^2-x^2}\,dx = \frac{1}{2A}\log\left|\frac{A+x}{A-x}\right|+C \]

6. More Special Forms

\[ \int \frac{1}{\sqrt{A^2-x^2}}\,dx = \sin^{-1}\left(\frac{x}{A}\right)+C \]

\[ \int \frac{1}{\sqrt{x^2-A^2}}\,dx = \log\left|x+\sqrt{x^2-A^2}\right|+C \]

\[ \int \frac{1}{\sqrt{A^2+x^2}}\,dx = \log\left|x+\sqrt{A^2+x^2}\right|+C \]

7. Root Integrals

\[ \int \sqrt{A^2-x^2}\,dx = \frac{x}{2}\sqrt{A^2-x^2} + \frac{A^2}{2}\sin^{-1}\left(\frac{x}{A}\right)+C \]

\[ \int \sqrt{x^2-A^2}\,dx = \frac{x}{2}\sqrt{x^2-A^2} - \frac{A^2}{2}\log\left|x+\sqrt{x^2-A^2}\right|+C \]

\[ \int \sqrt{A^2+x^2}\,dx = \frac{x}{2}\sqrt{A^2+x^2} + \frac{A^2}{2}\log\left|x+\sqrt{A^2+x^2}\right|+C \]

Common Mistakes

Wrong idea: Integral always means area.

Correct idea: Integral gives signed accumulation. Total area may require absolute value.

Wrong:

\[ \int \frac{1}{x}\,dx=\frac{x^0}{0} \]

Correct: