Integration - Teach.sg

Integration Rules

$\,\int k\,\text{d}x = kx +c$

$\,\int x^n\,\text{d}x = \dfrac{x^{n+1}}{n+1} +c, n\ne -1$

$\,\int \sin x\,\text{d}x = -\cos x +c$

$\,\int \cos x\,\text{d}x = \sin x +c$

$\,\int \sec^2 x\,\text{d}x = \tan x +c$

$\,\int e^x\,\text{d}x = e^x +c$

$\,\int \frac{1}{x}\,\text{d}x = \ln x +c$

Note: $\int k\text{f}(x)\,\text{d}x=k\times\int \text{f}(x)\,\text{d}x$

Further Integration Rules

$\,\int (ax+b)^n\,\text{d}x = \dfrac{(ax+b)^{n+1}}{a(n+1)} +c, n \ne -1$

$\,\int \sin (ax+b)\,\text{d}x = -\dfrac{\cos (ax+b)}{a} +c$

$\,\int \cos (ax+b)\,\text{d}x = \dfrac{\sin (ax+b)}{a} +c$

$\,\int \sec^2 (ax+b)\,\text{d}x = \dfrac{\tan (ax+b)}{a} +c$

$\,\int e^{ax+b}\,\text{d}x = \dfrac{e^{ax+b}}{a} +c$

$\,\int \frac{1}{ax+b}\,\text{d}x = \dfrac{\ln (ax+b)}{a}+c$

Definite Integral

For $\int \text{f}(x) \,\text{d}x = \text{F}(x) + c$,

$\displaystyle\int_a^b \text{f}(x) \,\text{d}x = \text{F}(b) -\text{F}(a)$

Area With Respect To $x$-axis Or $y$-axis

$\text{For area with respect to } x\text{-axis, } \displaystyle\int_a^b \text{f}(x) \,\text{d}x$

$\text{For area with respect to } y\text{-axis, } \displaystyle\int_c^d \text{f}(y) \,\text{d}y$

Note: For area below the $x$-axis (taken with respect to $x$-axis) or area to the left of the $y$-axis (taken with respect to $y$-axis), it is taken as negative.

Kinematics

$\,v=\dfrac{\text{d}s}{\text{d}t}$

$\,a=\dfrac{\text{d}v}{\text{d}t}$

$\,s=\displaystyle\int v\,\text{d}t$

$\,v=\displaystyle\int a\,\text{d}t$