Question

If $u_n={\int }_0^{\pi /2}{x}^n\sin xdx$, then the value of $u_{10}+90u_8$ is?

• A. $9{\left(\frac{\pi }{2}\right)}^8$
• B. ${\left(\frac{\pi }{2}\right)}^9$
• C. $10{\left(\frac{\pi }{2}\right)}^9$
• D. $9{\left(\frac{\pi }{2}\right)}^9$

Answer 1

The correct option is C $10{\left(\frac{\pi }{2}\right)}^9$

$4_{\pi }={\int }_0^{\frac{\pi }{2}}{x}^n\sin xdx$

$4_{10}={\int }_0^{\frac{\pi }{2}}{x}^{10}\sin xdx$

$4_{10}=x^{10}{\int }_0^{\frac{\pi }{2}}\sin x-{\int }_0^{\frac{\pi }{2}}10x^9\sin xdx$

$4_{10}=x^{10}(-\cos x{)}_0^{\frac{\pi }{2}}+10(x^9\sin x{)}_0^{\frac{\pi }{2}}-90{\int }_0^{\frac{\pi }{2}}{x}^8\sin x$

$4_{10}+{904}_8={\left(\frac{\pi }{2}\right)}^{10}\left(0\right)+10{\left(\frac{\pi }{2}\right)}^9$

$4_{10}+{904}_8=10{\left(\frac{\pi }{2}\right)}^9$.