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_(-5)^5 | x + 2| dx`
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
Evaluate`int (1)/(x(3+log x))dx`
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
`int_1^2 1/(2x + 3) dx` = ______
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
The value of `int_2^7 (sqrtx)/(sqrt(9 - x) + sqrtx)dx` is ______
`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Evaluate: `int_(-1)^3 |x^3 - x|dx`
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
Evaluate: `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x)dx`
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
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 ______.
`int_0^1|3x - 1|dx` equals ______.
If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is ______.
If `int_0^(π/2) log cos x dx = π/2 log(1/2)`, then `int_0^(π/2) log sec dx` = ______.
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is ______.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Evaluate: `int_-1^1 x^17.cos^4x dx`
Solve the following.
`int_0^1e^(x^2)x^3 dx`
Solve the following.
`int_2^3x/((x+2)(x+3))dx`
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
