Question

Prove the above :-

Want quality answers ✓

Don't Spam Please ✓

If you didn't get the answer . But you did work hard . So your answer will be marked as Brainliest ✓

​

Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:Let \: I_n \: = \: \displaystyle\int_{x=0}^\infty x^ne^{-x}\,dx$

Using By parts,

$\rm \: \: = \displaystyle \: -x^ne^{-x}\Big|_{x=0}^\infty+n\int_{x=0}^\infty x^{n-1}e^{-x}\,dx.$

$\rm \: \: = \displaystyle \: - \: (0 - 0) \: + \: n\int_{x=0}^\infty x^{n-1}e^{-x}\,dx.$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \: {e}^{ - \infty } = 0 \bigg \}}$

$\rm \: \: = \: 0 + nI_{n - 1}$

$\bf\implies \:I_n = n \: I_{n - 1} - - - (1)$

Consider,

$\rm :\longmapsto\: \: I_{n - 1} \: = \: \displaystyle\int_{x=0}^\infty x^{n - 1} e^{-x}\,dx$

On integrating by parts,

$\rm \: \: = \displaystyle \: -x^{n - 1}e^{-x}\Big|_{x=0}^\infty+(n - 1)\int_{x=0}^\infty x^{n-2}e^{-x}\,dx.$

$\rm \: \: = \displaystyle \: - \: (0 - 0) \: + \: (n - 1)\int_{x=0}^\infty x^{n-2}e^{-x}\,dx.$

$\rm \: \: = \: 0 + (n - 1)I_{n - 2}$

$\bf\implies \:I_{n - 1} = (n - 1)\: I_{n - 2} - - - (2)$

So, continuing like this ,

$\bf\implies \:I_{n - 2} = (n - 2)\: I_{n - 3} - - - (3)$

$\bf\implies \:I_{n - 3} = (n - 3)\: I_{n - 4} - - - (4)$

.

.

.

.

.

$\rm :\longmapsto\:\: I_ 0\: = \: \displaystyle\int_{x=0}^\infty e^{-x}\,dx$

$\rm \: \: = \:- \: e^{-x}\Big|_{x=0}^\infty$

$\rm \: \: = \: - ( {e}^{ - \infty } - {e}^{0})$

$\rm \: \: = \: - (0 - 1)$

$\rm \: \: = \: 1$

$\bf\implies \:I_0 = 1$

Hence,

Equation (1) can be rewritten as

$\rm :\longmapsto\:I_n = n(n - 1)(n - 2) - - - - 3 \times 2 \times I_0$

$\rm :\longmapsto\:I_n = n(n - 1)(n - 2) - - - - 3 \times 2 \times 1$

$\bf\implies \:I_n \: = \: n !$

$\bf\implies \:\: \displaystyle\int_{x=0}^\infty \bf \: x^ne^{-x}\,dx \: = \: n!$

Additional Information :-

Integration by Parts :-

✏️See the rule:

• ∫u v dx = u∫v dx −∫u' (∫v dx) dx

where,

• 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.