Question

find integration of
\int ( x ^ { 2 } - \frac { y } { 3 } + \frac { x ^ { 2 } - x ^ { 3 } } { x } ) d x​

Answer 1

Answer

$\displaystyle\int{ (x ^ { 2 } - \frac{ 4x }{ 3 } + \frac{ x ^ { 2 } -x ^ { 3 } }{ x } ) }d x \\ \int x^{2}-\frac{4x}{3}+\frac{\left(-x+1\right)x^{2}}{x}\mathrm{d}x \\ \int x^{2}-\frac{4x}{3}+x\left(-x+1\right)\mathrm{d}x \\ \int x^{2}-\frac{4x}{3}-x^{2}+x\mathrm{d}x \\ \int -\frac{4x}{3}+x\mathrm{d}x \\ \int -\frac{4x}{3}+\frac{3x}{3}\mathrm{d}x \\ \int \frac{-x}{3}\mathrm{d}x \\ -\frac{1}{3}\int x\mathrm{d}x \\ -\frac{1}{3}\times \left(\frac{x^{2}}{2}\right) \\ -\frac{x^{2}}{6} \\ \\ -\frac{x^{2}}{6}+С$

Answer 2

Solution:

Given Integral:

$\displaystyle \rm \longrightarrow I = \int \bigg[ {x}^{2} - \dfrac{y}{3} + \dfrac{ {x}^{2} - {x}^{3} }{x} \bigg] \: dx$

Can be written as:

$\displaystyle \rm \longrightarrow I = \int \bigg[ {x}^{2} - \dfrac{y}{3} +x - {x}^{2} \bigg] \: dx$

$\displaystyle \rm \longrightarrow I = \int \bigg[x- \dfrac{y}{3} \bigg] \: dx$

Can be written as:

$\displaystyle \rm \longrightarrow I = \int x \: dx- \int \dfrac{y}{3} \: dx$

$\displaystyle \rm \longrightarrow I = \int x \: dx- \dfrac{1}{3} \int y\: dx$

We know that:

$\displaystyle \rm \longrightarrow \int {x}^{n} \: dx = \dfrac{ {x}^{n + 1} }{n + 1} + C$

$\displaystyle \rm \longrightarrow \int y\: dx = yx +C$

Using this results, we get:

$\displaystyle \rm \longrightarrow I = \dfrac{ {x}^{2} }{2} - \dfrac{yx}{3} + C$

Therefore:

$\displaystyle \rm \longrightarrow \int \bigg[ {x}^{2} - \dfrac{y}{3} + \dfrac{ {x}^{2} - {x}^{3} }{x} \bigg] \: dx = \dfrac{ {x}^{2} }{2} - \dfrac{yx}{3} + C$

★ Which is our required answer.

Learn More:

$\boxed{\begin{array}{c|c}\bf f(x)&\bf\displaystyle\int\rm f(x)\:dx\\ \\ \frac{\qquad\qquad}{}&\frac{\qquad\qquad}{}\\ \rm k&\rm kx+C\\ \\ \rm sin(x)&\rm-cos(x)+C\\ \\ \rm cos(x)&\rm sin(x)+C\\ \\ \rm{sec}^{2}(x)&\rm tan(x)+C\\ \\ \rm{cosec}^{2}(x)&\rm-cot(x)+C\\ \\ \rm sec(x)\ tan(x)&\rm sec(x)+C\\ \\ \rm cosec(x)\ cot(x)&\rm-cosec(x)+C\\ \\ \rm tan(x)&\rm log(sec(x))+C\\ \\ \rm\dfrac{1}{x}&\rm log(x)+C\\ \\ \rm{e}^{x}&\rm{e}^{x}+C\\ \\ \rm x^{n},n\neq-1&\rm\dfrac{x^{n+1}}{n+1}+C\end{array}}$