Question

integrate 6x² – 2x+3 dx​

Answer 1

AnsWer :

2x³ - x² + 3x + c.

SolutioN :

Let's Integrate w.r.t.x

$\longmapsto \tt\: \: \int\Big(6 {x}^{2} - 2x + 3 \Big) \,dx$

• By Sum Rules,

$\longmapsto \tt\: \: \int6 {x}^{2}\,dx - \int 2x\,dx + \int 3\,dx$

• Taking Constant terms.

$\longmapsto \tt\: \: 6 \int {x}^{2}\,dx -2 \int x\,dx + 3\int \,dx$

$\longmapsto \tt\: \: 6 . \frac{ {x}^{3} }{3} -2 . \frac{ {x}^{2} }{2} + 3x + c$

$\longmapsto \tt\: \: \cancel6 . \frac{ {x}^{3} }{ \cancel3} - \cancel2 . \frac{ {x}^{2} }{ \cancel2} + 3x + c$

$\longmapsto \tt\: \: 2 {x}^{3} - {x}^{2} + 3x + c.$

Therefore, the value of integration of 6x² – 2x + 3 with respect to x is 2x³ - x² + 3x + c.

__________________________

MorE InformatioN :

$\tt \bullet \: \: \: \: \:\int {a}^{n}\,da = \dfrac{ {a}^{n + 1} }{n + 1} + c.$

$\tt \bullet \: \: \: \: \:\int \frac{1}{a} \,da = log\,a + c.$

$\tt \bullet \: \: \: \: \:\int 1 \,da = a + c.$

Answer 2

Question :

Integrate 6x² – 2x+3 dx​

Answer :

$\bf{2x^3-x^2+3x+C}$

Explanation :

$\bf{\int \:6x^2-2x+3dx=2x^3-x^2+3x+C}$

                                                                         

$\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx$

$=\int \:6x^2dx-\int \:2xdx+\int \:3dx$

                                               

$\int \:6x^2dx=2x^3$

           

$\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx$

$=6\cdot \int \:x^2dx$

$\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1$

$=6\cdot \frac{x^{2+1}}{2+1}$

                                               

$\int \:2xdx=x^2$

           

$\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx$

$=2\cdot \int \:xdx$

$\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1$

$=2\cdot \frac{x^{1+1}}{1+1}$

                                         

$\int \:3dx=3x$

           

$\mathrm{Integral\:of\:a\:constant}:\quad \int adx=ax$

$=3x$

                                               

$\bf{=2x^3-x^2+3x}$

$\mathrm{Add\:a\:constant\:to\:the\:solution}$

$\bf{=2x^3-x^2+3x+C}$

                                                                         

Regards :)