Question

$\huge\bf{Question}$ :- What is integration ? How we do it ? Give examples? and Give Some basic formulaes??? Only for mods &stars .... All the best :)​

Answer 1

Integration

If f and F are function of 'x' such that F'(x) = f(x) then the function F is called a Primitive or Anti - Derivative or Integral of f(x) with respect to 'x' and is written symbolically as :

$\sf {\displaystyle \int f(x)dx=F(x)+C}$

where

C is constant of Integration.

Geometrically, an integration represents a family of curve as different values of 'C' will correspond to different members of the family and these members can be obtained by shifting any one of the curve parallel to itself.

For example,

$\sf{f(x) = 2x\:\:\:then\:\displaystyle \int f(x)dx=x^2+C}$

For different values of C, we will get different members of the curve which is parabola in the stated case.

If integration is given some limits like :

$\sf {\displaystyle \int_{a}^b{} f(x)dx=F(b)-F(a)}$

then the integration will represent the algebraic area bounded by the curve f(x).

Generally, we can say that Integration stands for Summation of small - small components of the given function.

Solving Integration

We can solve integration mainly with the manipulation of given function such that it resembles to our given formulae. After manipulation, we can apply the basic formulae and properties.

Some basic techniques which we use are mentioned below :

• Substitution Method
• Integration by parts
• Partial Fraction Method

Example :-

Solve the following :

$\sf {i)\:\displaystyle \int sinxcosxdx}$

$\sf {ii)\: \displaystyle \int_{0}^{\pi/2} sinxcosxdx}$

Solution :-

i) Substitute, sinx = t

Differentiate both sides,

cosxdx = dt

Now, our question changes to,

$\sf {\displaystyle \int tdt}$

from basic formula, we can write it as,

$\sf {\dfrac{t^2}{2}+C}$

Now, putting back the value of t we get,

$\sf {\dfrac{sin^2x}{2}+C}$

ii) The question is same but we have to apply the limits.

So, taking answer from first part and applying limits on it,

$\sf {\left [ \dfrac{sin^2x}{2}+C \right ]_0^{\pi/2}}$

$\sf {\left [ \dfrac{sin^2{(\pi/2)}}{2}+C - \dfrac{sin^2(0)}{2}-C \right ]}$

Now,

$\sf {sin(\pi/2)=1\:\:and\:\:sin(0)=0}$

We can write,

$\sf {\left [ \dfrac{1^2}{2}+C - \dfrac{0}{2}-C \right ]}$

which equals to

$\sf {\dfrac{1}{2}}$

Basic Formulae

$\boxed{\boxed{\begin{minipage}{4cm}\displaystyle\circ\sf\:\int{1\:dx}=x+c\\\\\circ\sf\:\int{a\:dx}=ax+c\\\\\circ\sf\:\int{x^n\:dx}=\dfrac{x^{n+1}}{n+1}+c\\\\\circ\sf\:\int{sin\:x\:dx}=-cos\:x+c\\\\\circ\sf\:\int{cos\:x\:dx}=sin\:x+c\\\\\circ\sf\:\int{sec^2x\:dx}=tan\:x+c\\\\\circ\sf\:\int{e^x\:dx}=e^x+c\end{minipage}}}$

Answer 2

If f and F are function of 'x' such that F'(x) = f(x) then the function F is called a Primitive or Anti - Derivative or Integral of f(x) with respect to 'x' and is written symbolically as :

$\bold {\underline {\green{List \: of \: Integral \: Formulas}}}$

∫ 1 dx = x + C.

∫ a dx = ax+ C.

∫ xn dx = ((xn+1)/(n+1))+C ; n≠1.

∫ sin x dx = – cos x + C.

∫ cos x dx = sin x + C.

∫ sec2 dx = tan x + C.

∫ csc2 dx = -cot x + C.

∫ sec x (tan x) dx = sec x + C.