Question

Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`

Answer 1

Let I = `int xsqrt(5 - 4x - x^2).dx`

Let x = `"A"[d/dx(5 - 4x - x^2)] + "B"`

= A [– 4 – 2x] + B
= –2Ax + (B – 4A)
Comparing the coefficients of x and the constant term on both the sides, we get
–2A = 1, B – 4A = 0

∴  A = `-(1)/(2), "B" = 4"A" = 4(-1/2)` = – 2

∴ x = `-(1)/(2)(- 4 - 2x) - 2`

∴ I = `int [ -1/2 (- 4 - 2x) - 2]sqrt(5 - 4x - x^2).dx`

= `-(1)/(2) int (- 4 - 2x) sqrt(5 - 4x - x^2).dx - 2 int sqrt(5 - 4x - x^2).dx`

= I₁ - I₂
In I₁, put 5 - 4x - x2 = t
∴ (– 4 – 2x).dx = dt

∴ I₁ = `(1)/(2)int t^(1/2).dt `

= `-(1)/(2)(t^(3/2)/(3/2)) + c_1`

= `-(1)/(3)(5 - 4x - x^2)^(3/2) + c_1`

I₂ = `2 int sqrt(5 - 4x - x^2).dx`

= `2 int sqrt(5 - (x^2 + 4x)).dx`

= `2 int sqrt(9 - (x^2 + 4x + 4)).dx`

= `2 int sqrt(3^2 - (x + 2)^2).dx`

= `2[((x + 2)/2) sqrt(3^2 - (x + 2)^2) + 3^2/(2)sin^-1 ((x + 2)/3)] + c_2`

= `(x + 2)sqrt(5 - 4x - x^2) + 9sin^-1 ((x + 2)/3) + c_2`

∴ I = `-(1)/(3)(5 - 4x - x^2)^(3/2) - (x + 2) sqrt(5 - 4x - x^2) - 9sin^-1 ((x + 2)/3) + c`, where c = c₁ + c_{2 .}