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प्रश्न
Evaluate the following:
`int ("d"x)/(xsqrt(x^4 - 1))` (Hint: Put x2 = sec θ)
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उत्तर
Let I = `int ("d"x)/(xsqrt(x^4 - 1))`
= `int (x"d"x)/(x^2sqrt(x^4 - 1))`
Put x2 = sec θ
∴ 2x dx = sec θ tan θ dθ
x dx = `1/2 sec theta tan theta "d"theta`
∴ I = `1/2 int (sec theta tan theta)/(sec theta sqrt(sec^2theta - 1)) "d"theta`
= `1/2 int (sectheta tan theta)/(sectheta * tan theta) "d"theta`
= `1/2 int 1 "d"theta`
= `1/2 theta + "C"`
So I = `1/2 sec^-1x^2 + "C"`
Hence, I = `1/2 sec^-1 x^2 + "C"`.
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