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Question
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
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Solution
Let I = `int (1)/(x^3 sqrt(x^2 - 1))*dx`
Put x = secθ
∴ dx secθ tanθ dθ
∴ I = `int (secθ tanθ dθ)/(sec3θ sqrt(sec^2θ - 1)`
= `int (secθ tanθ dθ)/(sec^3θ sqrt(tan^2θ))*dθ`
∴ I = `int cos^2 θ *dθ`
= `(1)/(2) int (1 + cos 2θ)*dθ`
= `(1)/(2) int dθ + 1/2 int cos 2θ*dθ`
= `θ/(2) + (1)/(2)((sin2θ)/2) + c` ...(1)
∴ x = sec θ
∴ θ = sec–1x
sin2θ = 2 sinθ cos θ
= `2sqrt(1 - cos^2θ)*cosθ`
= `2sqrt(1 - 1/x^2)(1/x) ...[because secθ = x ⇒ cosθ = 1/x]`
= `(2sqrt(x^2 - 1))/x^2`
∴ from (1), we have
I = `(1)/(2)sec^-1 x + 1/2 sqrt(x^2 - 1)/x^2 + c`.
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