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प्रश्न
Evaluate the following:
`int_0^(1/sqrt(2)) ("e"^(sin^-1x) sin^-1 x)/sqrt(1 - x^2) "d"x`
योग
उत्तर
Let I = `int_0^(1/sqrt(2)) ("e"^(sin^-1x) sin^-1 x)/sqrt(1 - x^2) "d"x` .........(1)
Put t = `sin^-1 x` ........(2)
Differentiate with respect to x
`"dt"/("d"x) = 1/sqrt(1 - x^2)`
dt = `1/sqrt(1 - x^2)` .........(3)
| x | 0 | `1/sqrt(2)` |
| t | 0 | `pi/4` |
Substitute (2) and (3) in (1),
(1) ⇒ `int_0^(pi/4) "e"^"t" "t" "dt"`
Alplying Bernoulli's formula, we get
= `("e"^"t"["t" - 1])_0^(pi/4)`
=`"e"^(i/4)[pi/4 - 1] - "e"^0[0 - 1]`
= `"e"^(pi/4)[pi/4 - 1] + 1`
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Bernoulli’s Formula
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