# Integration

Integration Rules
$1.\,\int k\,\text{d}x = kx +c$
$2.\,\int x^n\,\text{d}x = \dfrac{x^{n+1}}{n+1} +c, n\ne -1$
$3.\,\int \sin x\,\text{d}x = -\cos x +c$
$4.\,\int \cos x\,\text{d}x = \sin x +c$
$5.\,\int \sec^2 x\,\text{d}x = \tan x +c$
$6.\,\int e^x\,\text{d}x = e^x +c$
$7.\,\int \frac{1}{x}\,\text{d}x = \ln x +c$
Note: $\int k\text{f}(x)\,\text{d}x=k\times\int \text{f}(x)\,\text{d}x$
Further Integration Rules
$1.\,\int (ax+b)^n\,\text{d}x = \dfrac{(ax+b)^{n+1}}{a(n+1)} +c, n \ne -1\\$
$2.\,\int \sin (ax+b)\,\text{d}x = -\dfrac{\cos (ax+b)}{a} +c\\$
$3.\,\int \cos (ax+b)\,\text{d}x = \dfrac{\sin (ax+b)}{a} +c\\$
$4.\,\int \sec^2 (ax+b)\,\text{d}x = \dfrac{\tan (ax+b)}{a} +c\\$
$5.\,\int e^{ax+b}\,\text{d}x = \dfrac{e^{ax+b}}{a} +c\\$
$6.\,\int \frac{1}{ax+b}\,\text{d}x = \dfrac{\ln (ax+b)}{a}+c$
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
For area with respect to $x$-axis, $\displaystyle\int_a^b \text{f}(x) \,\text{d}x$.
For area with respect to $y$-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
$1.\,v=\dfrac{\text{d}s}{\text{d}t}\\$
$2.\,a=\dfrac{\text{d}v}{\text{d}t}\\$
$3.\,s=\displaystyle\int v\,\text{d}t\\$
$4.\,v=\displaystyle\int a\,\text{d}t$

Note:
a. velocity, $v$ determines both the speed and the direction
b. $\text{average speed}=\dfrac{\text{total distance}}{\text{total time}}$
c. particle starts from origin, $s = 0$
d. instantaneously at rest, $v = 0$
e. max / min velocity, $a=\dfrac{\text{d}v}{\text{d}t}=0$
f. max / min displacement, $v=\dfrac{\text{d}s}{\text{d}t}=0$

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