Question

can anyone solve this ...help if you can { no spam wrong answer must be reported } chapter 0 physics class 11th Integrate the following :-​

Answer 1

Answer:

$\\\tt \dfrac{x^{3}}{3} + x + C$

Explanation:

Steps:

• Get the common terms in the numerator.
• Eliminate with denominator.
• Integrate.

Shown as:

$\\\tt \int \limits \dfrac{x^{3} - x^{2} + x -1 }{x-1} dx \\ \\\tt = \int \limits \dfrac{x^{2}(x - 1) +( x -1) }{x-1} dx \\ \\\tt = \int \limits \dfrac{(x^{2}+ 1) ( x -1) }{x-1} dx \\ \\\tt = \int \limits (x^{2} +1) dx \\ \\\tt = \dfrac{x^{3}}{3} + x + C$

Answer 2

Given : $\sf \int \dfrac{ x^3 - x^2 + x - 1}{ x - 1} dx \:$

Exigency To Solve : $\sf \int \dfrac{ x^3 - x^2 + x - 1}{ x - 1} dx \:$

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$\qquad \dag\:\:\bigg\lgroup \sf{ \int \dfrac{ x^3 - x^2 + x - 1}{ x - 1} dx }\bigg\rgroup$⠀⠀⠀

⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Solving \: the \: Given\::}}$

$\qquad:\implies \sf \int \dfrac{ x^3 - x^2 + x - 1}{ x - 1} dx$

$\qquad:\implies \sf \int \dfrac{ x^2 ( x - 1) + x - 1}{ x - 1} dx$

$\qquad:\implies \sf \int \dfrac{ ( x ^2 + 1) (x - 1)}{ x - 1} dx$

$\qquad:\implies \sf \int \dfrac{ ( x ^2 + 1) (x - 1)}{ (x - 1)} dx$

$\qquad:\implies \sf \int \dfrac{ ( x ^2 + 1) \cancel{(x - 1)}}{\cancel {(x - 1)}} dx$

$\qquad:\implies \sf \int (x^2 + 1 )dx$

$\qquad:\implies \sf \int (x^2 + x^0) dx$

$\qquad:\implies \sf \int x^2 dx + \int x^0 dx$

$\dag\:\:\sf{ As,\:We\:know\:that\::}$

$\qquad \dag\:\:\Bigg\lgroup \sf{ \int x^a dx = \dfrac{x^{a + 1}}{a + 1} }\Bigg\rgroup$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Applying\: \: this \::}}$

$\qquad:\implies \sf \int x^2 dx + \int x^0 dx$

$\dag\:\:\sf{ As,\:We\:know\:that\::}$

$\qquad \dag\:\:\Bigg\lgroup \sf{ \int x^a dx = \dfrac{x^{a + 1}}{a + 1} }\Bigg\rgroup$⠀

⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Applying\: \: this \::}}$

$\qquad:\implies \sf \int x^2 dx + \int x^0 dx$

$\qquad:\implies \sf \dfrac{x^{2 + 1}}{2 + 1} + \dfrac{x^{0 + 1}}{0 + 1} + c$

$\qquad:\implies \sf \dfrac{x^{2 + 1}}{2 + 1} + \dfrac{x^{1}}{0 + 1} + c$

$\qquad:\implies \sf \dfrac{x^{2 + 1}}{2 + 1} + \dfrac{x^{1}}{ 1} + c$

$\qquad:\implies \sf \dfrac{x^{2 + 1}}{2 + 1} + \dfrac{x}{ 1} + c$

$\qquad:\implies \sf \dfrac{x^{2 + 1}}{2 + 1} + x + c$$\qquad:\implies \sf \dfrac{x^{3}}{2 + 1} + x + c$

$\qquad :\implies \frak{\underline{\purple{\:Answer \: =\: \dfrac{x^{3}}{3} + x + c \: }} }\:\:\bigstar$

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline{ \mathrm {\: \:Answer \:is\:\:\:\:\bf{ \dfrac{x^{3}}{3} + x + c }}}}$

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