Advertisements
Advertisements
प्रश्न
Solve the following : `int_1^2 (5x^2)/(x^2 + 4x + 3)*dx`
उत्तर
Let I = `int_1^2 (5x^2)/(x^2 + 4x + 3)*dx`
= `5 int_1^2 x^2/(x^2 + 4x + 3)*dx`
Dividing numerator by denominator, we get
1
`x^2 + 4x + 3)x^2`
x2 + 4x + 3
– – –
– 4x – 3
∴ I = `5 int_1^2(1 - (4x + 3)/(x^2 + 4x + 3))*dx`
= `5 int_1^2 1*dx - 5 int_1^2 (4x + 3)/(x^2 + 4x + 3)*dx`
= `5 int_1^2 1*dx - 5 int_1^2 (4x + 3)/((x + 3)(x + 1))*dx`
Let `(4x + 3)/((x + 3)(x + 1)) = "A"/(x + 3) + "B"/(x + 1)` ...(i)
∴ 4x + 3 =A(x + 1) + B(x + 3) ...(ii)
Putting x = – 1 in (ii), we get
– 4 + 3 = A (– 1 + 1) + B(– 1 + 3)
∴ – 1 = 2B
∴ B = `-(1)/(2)`
Putting x = –3 in (ii), we get
– 12 + 3 = A(– 3 + 1) + B(– 3 + 3)
∴ – 9 = – 2A
∴ A = `(9)/(2)`
From (i), we get
`(4x + 3)/((x + 3(x + 1))`
= `(9/2)/(x + 3) + ((-1/2))/(x + 1)`
∴ I = `5 int_1^2 1*dx - 5 int_1^2[(9/2)/(x + 3) + ((-1/2))/(x + 1)]*dx`
= `5[x]_1^2 - 5[9/2 int_1^2 (1)/(x + 3)*dx - (1)/(2) int_1^2 (1)/(x + 1)*dx]`
= `5 (2 - 1) - 5{9/2 [log |x + 3|]_1^2 - (1)/(2)[log|x + 1|]_1^2}`
= `5 - 5[9/2(log 5 - log4) - (1)/(2)(log 3 - log 2)]`
= `5 - (5)/(2)[9 (log 5 - log2^2) - (log 3 - log 2)]`
= `5 - (5)/(2) [9 (log 5 - 2 log 2) log 3 + log 2]`
= `5 - (5)/(2) (- log 3 - 17 log 2 + 9 log 5)`
∴ I = `5 + (1)/(2) (5 log 3 + 85 log 2 - 45 log 5)`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `int_3^5 (1)/(sqrt(2x + 3) - sqrt(2x - 3))*dx`
Evaluate:
`int_0^(pi/4) sqrt(1 + sin 2x)*dx`
Evaluate : `int_0^(pi/4) (sec^2x)/(3tan^2x + 4tan x +1)*dx`
Evaluate : `int_0^(pi/4) (cosx)/(4 - sin^2x)*dx`
Evaluate the following : `int_(pi/5)^((3pi)/10) sinx/(sinx + cosx)*dx`
Evaluate the following : `int_0^1 sin^-1 ((2x)/(1 + x^2))*dx`
Evaluate the following : `int_0^pi (sin^-1x + cos^-1x)^3 sin^3x*dx`
Choose the correct alternative :
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =
Solve the following : `int_(-2)^3 (1)/(x + 5)*dx`
`int_1^2 x^2 "d"x` = ______
State whether the following statement is True or False:
`int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x) "d"x`
`int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2))` dx = ?
Evaluate the following definite intergral:
`int_1^2 (3x)/((9x^2 - 1))dx`
`int_0^4 1/sqrt(4x - x^2)dx` = ______.
Evaluate the following definite intergral:
`int_4^9 1/sqrt(x)dx`
Solve the following.
`int_1^3 x^2 log x dx`
Evaluate the following definite intergral:
`int_-2^3 1/(x+5) · dx`
Solve the following.
`int_1^3x^2 logx dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`