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प्रश्न
Evaluate the integral by using substitution.
`int_0^1 sin^(-1) ((2x)/(1+ x^2)) dx`
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उत्तर
Let `int_0^1 sin^-1 ((2x)/(1 + x^2)) dx`
Substituting x = tan θ
`dx = sec^2 theta d theta`
And `(2 tan theta)/(1 + tan^2 theta) = sin 2 theta`
When x = 0
⇒ θ = 0
or x = 1
`=> theta = pi/4`
Hence, `int_0^(pi/4) sin^-1 (sin 2 theta) xx sec^2 theta d theta`
`2 = int_0^(pi/4) theta sec^2 theta d theta`
`= 2 [(theta . tan theta)_0^(pi/4) - int_0^(pi/4) 1 * tan theta d theta]`
`= 2 [pi/4 tan pi/4 - 0] - 2 [log cos theta]_0^(pi/4)`
`= pi/4 + 2 [log cos pi/4 - log cos 0]`
`= pi/2 + 2 [log 1/sqrt2 - log 1]`
`= pi/2 - log 2`
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