Advertisements
Advertisements
Question
Find : `int_ (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.
Advertisements
Solution
Let `I = int_ (2x+1)/((x^2+1)(x^2+4))dx`
Let `(2x+1)/((x^2+1)(x^2+4)) = (Ax + B)/(x^2 + 1) + (Cx + D)/(x^2 + 4)`
Getting A = `2/3, B = 1/3, C = (-2)/3, D = (-1)/3`
∴ `I = 2/3 int x/(x^2 + 1) dx + 1/3 int x/(x^2 + 1)dx + (- 2)/3 int (xdx)/(x^2 + 4) + (-1)/3 int dx/(x^2 + 4)`
= `1/3 log | x^2 + 1| + 1/3 tan^-1 x - 1/3 log | x^2 + 4| - 1/6 tan^-1 x/2 + C`.
APPEARS IN
RELATED QUESTIONS
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
By using the properties of the definite integral, evaluate the integral:
`int_((-pi)/2)^(pi/2) sin^2 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^4 |x - 1| dx`
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
Using properties of definite integrals, evaluate
`int_0^(π/2) sqrt(sin x )/ (sqrtsin x + sqrtcos x)dx`
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
`int_1^2 1/(2x + 3) dx` = ______
`int_(-7)^7 x^3/(x^2 + 7) "d"x` = ______
The value of `int_-3^3 ("a"x^5 + "b"x^3 + "c"x + "k")"dx"`, where a, b, c, k are constants, depends only on ______.
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_0^{pi/2} cos^2x dx` = ______
`int_0^1 x tan^-1x dx` = ______
`int_3^9 x^3/((12 - x)^3 + x^3)` dx = ______
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
`int_0^pi x*sin x*cos^4x "d"x` = ______.
`int_0^9 1/(1 + sqrtx)` dx = ______
`int_(-1)^1 (x + x^3)/(9 - x^2) "d"x` = ______.
Evaluate `int_(-1)^2 "f"(x) "d"x`, where f(x) = |x + 1| + |x| + |x – 1|
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to ______.
`int (dx)/(e^x + e^(-x))` is equal to ______.
The value of `int_((-1)/sqrt(2))^(1/sqrt(2)) (((x + 1)/(x - 1))^2 + ((x - 1)/(x + 1))^2 - 2)^(1/2)`dx is ______.
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
Solve the following.
`int_1^3 x^2 logx dx`
Evaluate `int_0^3root3(x+4)/(root3(x+4)+root3(7-x)) dx`
Evaluate the following integral:
`int_0^1x (1 - x)^5 dx`
Solve the following.
`int_2^3x/((x+2)(x+3))dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
The value of \[\int_{-1}^{1}\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right)\mathrm{d}x\] is
