# Integrate the following w.r.t.x : cot–1 (1 – x + x2) - Mathematics and Statistics

Sum

Integrate the following w.r.t.x : cot–1 (1 – x + x2)

#### Solution

Let I = int cot^-1 (1 - x + x^2)*dx

= int tan^-1 (1/(1 - x + x^2))*dx

= int tan^-1 [(x + (1 - x))/(1 - x(1 - x))]

= int [tan^-1 x + tan^-1 (1 - x)]*dx

= int tan^-1 x*dx + int tan^-1 (1 - x)*dx

∴ I = I1 + I2                                                 ...(1)

I1 = int tan^-1 x*dx = int(tan^-1x)1*dx

= (tan^-1x)* int 1dx - [d/dx (tan^-1x)* int 1dx]*dx

= (tan^-1x)x - int 1/(1 + x^2)*x*dx

= xtan^-1 x - (1)/(2) int (2x)/(1 + x^2)*dx

∴ I1 = x tan^-1x - (1)/(2)log|1 + x^2| + c_1

...[because d/dx (1 + x^2) = 2x and int (f'(x))/f(x) dx = log|f(x)| + c]

I2 = int tan^-1 (1 - x)*dx

= int tan^-1 (1 - x)]*1dx

= [tan^-1 (1 - x)]*int 1dx - int {d/dx [tan^-1 (1 - x)]* int 1dx}*dx

= [tan^-1 (1 - x)]*x - int (1)/(1 + (1 - x)^2)*(-1)*xdx

= xtan^-1 (1 - x) + int x/(1 + 1 - 2x + x^2)*dx

= xtan^-1 (1 - x) + int x/(2 - 2x + x^2)*dx

Let x = "A"[d/dx (2 - 2x + x^2)] + "B"

∴ x = A(– 2 + 2x) + B = 2Ax + (–2A + B)
Comparing the coefficient of x and constant on both the sides, we get
1 = 2A and 0 = – 2A + B

∴ A = (1)/(2) and 0 = -2(1/2) + "B"

∴ B = 1

∴ x = (1)/(2)(- 2 + 2x) + 1

∴ I2= xtan^-1 (1 - x) + int (1/2(-2 + 2x) + 1)/(2 - 2x + x^2)*dx

= xtan^-1 (1 - x) + 1/2 (-2 + 2x)/(2 - 2x + x^2)*dx + int (1)/(2 - 2x + x^2)*dx

= xtan^-1 (1 - x) + (1)/(2) log|2 - 2x + x^2| + int (1)/(1 + (1 - 2x + x^2))*dx

= xtan^-1 (1 - x) + (1)/(2) log|x^2 - 2x + 2| + int (1)/(1 + (1 -  x^2))*dx

= xtan^-1 (1 - x) + (1)/(2) log|x^2 - 2x + 2| + (1)/(1) (tan-1 (1 - x))/(-1) + c_2

= x tan^-1 (1 - x) + 1/2log|x^2 - 2x + 2| - tan^-1 (1 - x) + c_2

= (x - 1)tan^-1 (1 - x) + (1)/(2)log|x^2 - 2x + 2| + c_2

∴ I2 = -(1 - x)tan^-1 (1 - x) + (1)/(2)log|x^2 - 2x + 2| + c_2             ...(3)

From (1),(2) and (3), we get

I = x tan^-1 x - (1)/(2) log|1 + x^2| + c_1 - (1 - x)tan^-1 (1 - x) + 1/2log|x^2 - 2x + 2| + c_2

= x tan^-1 x - (1)/(2) log|1 + x^2| - (1 - x)tan^-1 (1 - x) + 1/2 |x^2 - 2x + 2| + c, where c = c1 + c2.

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Chapter 3: Indefinite Integration - Miscellaneous Exercise 3 [Page 150]

#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 3 Indefinite Integration
Miscellaneous Exercise 3 | Q 3.02 | Page 150

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