Question

integral of (2x^2+2)dx​

Answer 1

Question :

Integrate

$\bf\int\:(2x^2+2)dx$

Theory :

•Properties of the indefinite integral

1) The indefinite integral of sum or difference of two function is equal to sum or difference of their integrals respectively.

$\sf\int[f_1(x)\pm\:f_2(x)]dx=\int\:f_1(x)dx\pm\int\:f_2(x)dx$

2) A constant factor of integrant may be taken outside the integral sign, if k is constant ,then

$\sf\int\:kf(x)dx=k\int\:f(x)dx$

Solution

We have to integrate :

$\rm\int\:(2x^2+2)dx$

We know that

$\sf\int[f_1(x)\pm\:f_2(x)]dx=\int\:f_1(x)dx\pm\int\:f_2(x)dx$

Then ,

$\sf\int\:(2x^2+2)dx=\int\:2x^2\:dx+\int2dx$

$\sf=2\int\:x^2\:dx+2\int\:x^{0}\:dx$

We know that

$\sf\int\:x^n=\dfrac{x{}^{n+1}}{n+1}$

Then ,

$\sf=2\times\dfrac{x^{2+1}}{2+1}+2\times\dfrac{(x^{0+1})}{(0+1)}+c$

$\sf=2\times\dfrac{x^{3}}{3}+2\dfrac{x^{1}}{1}+c$

$\sf=\dfrac{2x^3}{3}+2x+c$

$\sf=\dfrac{2x^3+6x}{3}+c$

Therefore,

$\rm\int\:(2x^2+2)dx=\dfrac{2}{3}x^3+2x+c$