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Integrate the following w.r.t.x : sec4x cosec2x - Mathematics and Statistics

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Sum

Integrate the following w.r.t.x : sec4x cosec2x

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Solution

Let I = `int sec^4x  "cosec"^2x*dx`

= `int sec^4x  "cosec"^2x* sec^2x*dx`

Put tan x = t
∴ sec2x·dx = d
Also, sec2x cosec2x = (1 + tan2x)(1 + cot2x)

= `(1 + t^2)(1 + 1/t^2)`

= `(1 + t^2)((t^2 + 1)/t^2)`

= `(t^4 + 2t^2 + 1)/t^2`

= `t^2 + 2 + (1)/t^2`

∴ I = `int (t^2 + 2 + 1/t^2)*dt`

= `int t^2*dt + 2 int *dt + int 1/t^2*dt`

= `t^3/(3) + 2t + (t^-1)/((-1)) + c`

= `(1)/(3)tan^3x + 2tanx - (1)/tanx + c`

= `(1)/(3cot^3x) + (2)/(cotx) - cot x + c`.

Concept: Methods of Integration: Integration by Parts
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 3 Indefinite Integration
Miscellaneous Exercise 3 | Q 3.2 | Page 150

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