∫π2/16π2/4 sin⁡xxdx=

Put x=t⇒12xdx=dtx=π216⇒t=π4,x=π24⇒t=π2∫π2/16π2/4 sin⁡(x)xdx=∫π/4π/2 [sin⁡t⋅2dt=−2cos⁡t]π/4π/2=−20−12=2

∫π2/16π2/4 sin⁡xxdx=

Put x=t⇒12xdx=dtx=π216⇒t=π4,x=π24⇒t=π2∫π2/16π2/4 sin⁡(x)xdx=∫π/4π/2 [sin⁡t⋅2dt=−2cos⁡t]π/4π/2=−20−12=2

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$\int_{π^{2} / 16}^{π^{2} / 4} \frac{\sin \sqrt{x}}{\sqrt{x}} dx =$

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$\sqrt{2}$

$1 / \sqrt{2}$

$2 \sqrt{2}$

$π / 2$

answer is A.

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Detailed Solution

Put $\sqrt{x} = t \Rightarrow \frac{1}{2 \sqrt{x}} dx = dt$
$\begin{array}{l} x = \frac{π^{2}}{16} \Rightarrow t = \frac{π}{4}, x = \frac{π^{2}}{4} \Rightarrow t = \frac{π}{2} \\ \int_{π^{2} / 16}^{π^{2} / 4} \frac{\sin (\sqrt{x})}{\sqrt{x}} dx = \int_{π / 4}^{π / 2} [\sin t \cdot 2 dt = - 2 \cos t]_{π / 4}^{π / 2} = - 2 [0 - \frac{1}{\sqrt{2}}] = \sqrt{2} \end{array}$