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Question
\[\int_0^\frac{\pi^2}{4} \frac{\sin\sqrt{x}}{\sqrt{x}} dx\] equals
Options
2
1
π/4
π2/8
Solution
2
\[\int_0^\frac{\pi^2}{4} \frac{\sin\sqrt{x}}{\sqrt{x}} d x\]
\[Let\ \sqrt{x} = t, then\ \frac{1}{2\sqrt{x}}dx = dt\]
\[When\ \ x = 0, t = 0, x = \frac{\pi^2}{4}, t = \frac{\pi}{2}\]
\[\text{Therefore the integral becomes}\]
\[ \int_0^\frac{\pi}{2} 2\ sint\ dt\]
\[ = - 2 \left[ cost \right]_0^\frac{\pi}{2} \]
\[ = 2\]
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