Integrate the following functions w.r.t. x : ∫13+2sinx-cosxdx - Mathematics and Statistics

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Sum

Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`

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Solution

Let I = `int (1)/(3 + 2sin x - cosx)dx`

Put `tan(x/2)` = t
∴ x = 2 tan–1 t

∴ dx = `(2)/(1 + t^2)dt` and

sinx = `(2t)/(1 + t^2)' cosx = (1 - t^2)/(1 + t^2)`

∴ I = `int (1)/(3 + 2((2t)/(1 + t^2)) - ((1 - t^2)/(1 + t^2))).(2dt)/(1 + t^2)`

= `int (1 + t^2)/(3(1 + t^2) + 4t - (1 - t^2)).(2dt)/(1 + t^2)`

= `2 int dt/(4t^2 + 4t + 2)`

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

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

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

= `tan^-1[2tan^-1(x/2) + 1] + c`.

  Is there an error in this question or solution?
Chapter 3: Indefinite Integration - Exercise 3.2 (B) [Page 123]

APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 3 Indefinite Integration
Exercise 3.2 (B) | Q 2.4 | Page 123

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