Question

evaluate integral x²e^xdx​

Answer 1

$\rm \longrightarrow \displaystyle \int \rm {x}^{2} {e}^{x} \: dx \\ \\ \\ \rm \longrightarrow {x}^{2}\displaystyle \int \rm {e}^{x} \: dx - \int \rm \bigg( \frac{d( {x}^{2} )}{dx} \int{e}^{x} \: dx \bigg) dx\\ \\ \\ \rm \longrightarrow {x}^{2}\rm {e}^{x} - \int \rm 2x \: {e}^{x} dx\\ \\ \\ \rm \longrightarrow {x}^{2}\rm {e}^{x} - 2\int \rm x \: {e}^{x} dx$

$\rm \longrightarrow {x}^{2}\rm {e}^{x} - 2\int \rm x \: {e}^{x} dx \\ \\ \\ \rm \longrightarrow {x}^{2}\rm {e}^{x} - 2 \bigg[x\int \rm \: {e}^{x} dx - \int \bigg( \frac{dx}{dx} \int {e}^{x} dx \bigg)dx\bigg]\\ \\ \\ \rm \longrightarrow {x}^{2}\rm {e}^{x} - 2 \bigg[x \rm \: {e}^{x} - \int {e}^{x} dx\bigg]\\ \\ \\ \rm \longrightarrow {x}^{2}\rm {e}^{x} - 2 (x \rm \: {e}^{x} - {e}^{x})+ c\\ \\ \\ \rm \longrightarrow {x}^{2}\rm {e}^{x} - 2 x \rm \: {e}^{x} - 2{e}^{x}+ c$

Answer 2

$\large\underline\blue{\bold{Given \:  Question :-  }}$

$\bf \:Evaluate : \bf\int\limits {x}^{2} {e}^{x} dx$

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$\large\underline\blue{\bold{Formula \: used:- }}$

✏️Integration by Parts

Integration by Parts is a special method of integration that is often useful when two functions are multiplied together, but is also helpful in other ways.

✏️See the rule:

∫u v dx = u∫v dx −∫u' (∫v dx) dx

• u is the function u(x)

• v is the function v(x)

• u' is the derivative of the function u(x)

For integration by parts , the ILATE rule is used to choose u and v.

where,

• I - Inverse trigonometric functions
• L -Logarithmic functions
• A - Arithmetic and Algebraic functions
• T - Trigonometric functions
• E- Exponential functions

The alphabet which comes first is choosen as u and other as v.

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$\huge {AηsωeR} ✍$

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$\bf\int\limits {x}^{2} {e}^{x} dx$

✏️Here,

$\sf \:  ⟼u = {x}^{2} \: and \: v \: = {e}^{x}$

$\underline{\boxed{\star{\sf{\blue{ Using \: Integration \: by \: Parts \: we \: get}}}}}$

$\sf \:  = {x}^{2} \sf\int\limits{e}^{x} dx - \bf\int\limits(\dfrac{d}{dx} {x}^{2} \sf\int\limits{e}^{x} dx)dx$

$\sf \:  = {x}^{2} {e}^{x} - \sf\int\limits2x{e}^{x} dx$

$\sf \:  = {x}^{2} {e}^{x} - 2\sf\int\limits \: x{e}^{x} dx$

$\underline{\boxed{\star{\sf{\blue{ Using \: Integration \: by \: Parts \: in \: second \: term \: we \: get}}}}}$

$\sf \:  = {x}^{2} {e}^{x} -2(x\sf\int\limits{e}^{x} dx - \sf\int\limits(\dfrac{d}{dx} x\sf\int\limits{e}^{x} dx)dx)$

$\sf \:  = {x}^{2} {e}^{x} -2x{e}^{x} + 2\sf\int\limits{e}^{x} dx$

$\sf \:  = {x}^{2} {e}^{x} -2x{e}^{x} +2{e}^{x} + c$

$\underline{\boxed{\star{\sf{\blue{ Hence, \bf\int\limits {x}^{2} {e}^{x} dx = \sf \:  = {x}^{2} {e}^{x} -2x{e}^{x} +2{e}^{x} + c}}}}}$

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