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11Alexandr11 [23.1K]
2 years ago
13

Find the definite integral from 0 to 1/16 of arcsin(8x)/sqrt(1-64x^2)dx

Mathematics
1 answer:
kolezko [41]2 years ago
8 0
\displaystyle\int_0^{1/16}\frac{\arcsin8x}{\sqrt{1-64x^2}}\,\mathrm dx

First let y=8x, so that \mathrm dx=\dfrac{\mathrm dy}8 to write the integral as

\displaystyle\frac18\int_0^{1/2}\frac{\arcsin y}{\sqrt{1-y^2}}\,\mathrm dy

Now recall that (\arcsin y)'=\dfrac1{\sqrt{1-y^2}}, so substituting z=\arcsin y should do the trick. The integral then becomes

\displaystyle\frac18\int_0^{\pi/6}z\,\mathrm dz=\frac1{16}z^2\bigg|_{z=0}^{z=\pi/6}=\frac{\pi^2}{576}
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