x ( )/(2)lim( _( /2)^3^x^3( (2t^1/3)+ (t^1/3))dt(x-( )/(2))^2) is equal to

Question

$\underset{x	o \frac{\pi }{2}}{lim}(\frac{{\int }_{(\pi /2{)}^{3}}^{{x}^{3}}(\sin (2{t}^{1/3})+\cos ({t}^{1/3}))dt}{{(x-\frac{\pi }{2})}^{2}})$ is equal to

Accepted Answer

$\underset{x	o \frac{\pi }{2}}{lim}(\frac{{\int }_{(\pi /2{)}^{3}}^{{x}^{3}}(\sin (2{t}^{1/3})+\cos ({t}^{1/3}))dt}{{(x-\frac{\pi }{2})}^{2}})$

Using Newton Leibniz theorem

$\begin{gathered} =\underset{x	o \frac{\pi }{2}}{lim}[\frac{\sin (2	imes \frac{\pi }{2})\cdot 0-\sin (2x)\cdot 3{x}^{2}+(\cos \frac{\pi }{2})\cdot 0-\cos x\cdot 3{x}^{2}}{2(x-\frac{\pi }{2})}] \ =\underset{w}{lim}x	o \frac{\pi }{2}\frac{-3{x}^{2}\sin 2x-3{x}^{2}\cos x}{2(x-\frac{\pi }{2})}(\frac{0}{0}) \ =\underset{x	o \frac{\pi }{2}}{lim}[\frac{-6x\sin 2x-6{x}^{2}\cos 2x-6x\cos x+3{x}^{2}\sin x}{2}] \ =\frac{6	imes \frac{{\pi }^{2}}{4}+3	imes \frac{{\pi }^{2}}{4}}{2} \ =\frac{9{\pi }^{2}}{8} \end{gathered}$

Solutions · 1