If-u-10-0-pi-2-x-10-sin-x-dx-then-the-value-of-u-10-90-u-8-is-

Question Number 41579 by Dawajan Nikmal last updated on 09/Aug/18

If u_{10} = \underset{0}{\int^{π / 2}} x^{10} \sin x d x, then the value of

u_{10} + 90 u_{8} is

Commented by maxmathsup by imad last updated on 11/Aug/18

l e t u_{n} = \int_{0}^{\frac{π}{2}} x^{n} s i n x d x l e t f i n d u_{n} b y r e c u r r e n c e b y p a r t s w e h a v e

u_{n} = {[\frac{1}{n + 1} x^{n + 1} s i n x]}_{0}^{\frac{π}{2}} - \int_{0}^{\frac{π}{2}} \frac{1}{n + 1} x^{n + 1} c o s x d x

= \frac{π^{n + 1}}{(n + 1) 2^{n + 1}} - \frac{1}{n + 1} \int_{0}^{\frac{π}{2}} x^{n + 1} c o s x d x

= \frac{π^{n + 1}}{(n + 1) 2^{n + 1}} - \frac{1}{(n + 1)} {{[\frac{1}{n + 2} x^{n + 2} c o s x]}_{0}^{\frac{π}{2}} + \int_{0}^{\frac{π}{2}} \frac{1}{(n + 2)} x^{n + 2} s i n x d x}

= \frac{π^{n + 1}}{(n + 1) 2^{n + 1}} - \frac{1}{(n + 1) (n + 2)} u_{n + 2} \Rightarrow

u_{8} = \frac{π^{9}}{9 . 2^{9}} - \frac{1}{9.10} u_{10} \Rightarrow u_{10} + 90 u_{8} = u_{10} + 90 \frac{π^{9}}{9 . 2^{9}} - u_{10} \Rightarrow

u_{10} + 90 u_{8} = \frac{10 \times π^{9}}{2^{9}} = 10 {(\frac{π}{2})}^{9} .

Answered by tanmay.chaudhury50@gmail.com last updated on 09/Aug/18

\int x^{10} s i n x d x

= x^{10} . (- c o s x) - \int 10 x^{9} . (- c o s x) d x

= - x^{10} c o s x + 10 \int x^{9} c o s x d x

= - x^{10} c o s x + 10 [(x^{9} s i n x - 9 \int x^{8} s i n x d x]

= - x^{10} c o s x + 10 x^{9} s i n x - 90 \int x^{8} s i n x d x

s o \int_{0}^{\frac{Π}{2}} x^{10} s i n x d x

=∣ - x^{10} c o s x + 10 x^{9} s i n x ∣_{0}^{\frac{Π}{2}} + (- 90) \int_{0}^{\frac{Π}{2}} x^{8} s i n x

u_{10} + 90 u_{8} = {- {(\frac{Π}{2})}^{10} c o s \frac{Π}{2} + 10 . {(\frac{Π}{2})}^{9} s i n \frac{Π}{2}}

u_{10} + 90 u_{8} = 10 . {(\frac{Π}{2})}^{9}