Question

find the intergation of (x2+2x)dx = x2dx +2(xdx​

Answer 1

$\\ \rm \int(x^2 + 2 x). dx \\ \\ \rm Integrate \: the \: sum \: term \: by \: term \: and \\ \rm factor \: out \: constants: \\ \rm \implies \int x^2. dx + 2 \int x .dx \\ \\ \rm By \: using \: power \: rule, \\ \rm \int {x}^{n} .dx = \dfrac{ {x}^{n + 1} }{n + 1} : \\ \rm \implies \dfrac{ {x}^{3} }{3} + \cancel{2} \times \dfrac{ {x}^{2} }{ \cancel{2}} + c \\ \\ \rm \implies \dfrac{ {x}^{3} }{3} + {x}^{2} + c$

Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\bf{\int \left(x^2+2x\right)dx=x^2dx+2\left(xdx\right)dx}$

♣ ᴀɴꜱᴡᴇʀ :

$\boxed{\bf{\int \:x^2+2xdx=\dfrac{x^3}{3}+x^2+C}}$

♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴ :

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

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

ꜱᴏʟᴠᴇ $\bf{\int \:x^2dx}$ :

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

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

$\sf{Simplify}$

$\sf{=\dfrac{x^3}{3}}$

ꜱᴏʟᴠᴇ $\bf{\int \:2xdx}$ :

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

$\sf{=2\cdot \int \:xdx}$

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

$\sf{=2\cdot \dfrac{x^{1+1}}{1+1}}$

ꜱɪᴍᴘʟɪꜰʏ $\sf{2\cdot \dfrac{x^{1+1}}{1+1}}$ :

$\sf{Add\:the\:numbers:}\:1+1=2}$

$\sf{=2\cdot \dfrac{x^2}{2}}$

$\sf{Multiply\:fractions}:\quad \:a\cdot \dfrac{b}{c}=\dfrac{a\:\cdot \:b}{c}}$

$\sf{=\dfrac{x^2\cdot \:2}{2}}$

$\sf{=x^2}$

$\bf{\int \:x^2dx+\int \:2xdx} = \bf{\dfrac{x^3}{3}+x^2}$

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

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