Advertisements
Advertisements
Question
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Advertisements
Solution
Let I = `int (2x + 1)/(x^2 + 4x - 5).dx`
Let 2x + 1 = `"A"[d/dx(x^2 + 4x - 5)] + "B"`
2x + 1 = A(2x + 4) + B
∴ 2x + 1 = 2Ax + (4A + B)
Comparing the coefficient of x and constant on both sides, we get,
| 2A = 2 | and | 4A + B = 1 |
| ∴ A = 1 | and | ∴ 4(1) + B = 1 |
| ∴ B = 1 - 4 | ||
| ∴ B = - 3 |
∴ 2x + 1 = (2x + 1) - 3
∴ I = `int ((2x + 1) - 3)/(x^2 + 4x + 5)."dx"`
∴ I = `int (2x + 1)/(x^2 + 4x - 5)."dx" - 3 int (1)/(x^2 + 4x - 5)."dx"`
∴ I = `"I"_1 - 3"I"_2`
I1 is of the type `int (f'(x))/f(x).dx = log|f(x)| + c`
∴ `"I"_1 = log|x^2 + 4x - 5| + c_1`
∴ I2 = `int (1)/(x^2 + 4x - 5).dx`
∴ I2 = `int (1)/((x^2 + 4x + 4) - 4 - 5).dx`
∴ I2 = `int (1)/((x + 2)^2 - 3^2).dx`
∴ I2 = `1/(2 × 3) log |(x + 2 - 3)/(x + 2 + 3)| + c_2`
∴ I2 = `1/6 log |(x - 1)/(x + 5)| + c_2`
∴ I = `log|x^2 + 4x - 5| - 3 × 1/6 log|(x - 1)/(x + 5)| + c`.
∴ I = `log|x^2 + 4x - 5| - 1/2 log|(x - 1)/(x + 5)| + c`.
APPEARS IN
RELATED QUESTIONS
Evaluate :`intxlogxdx`
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Integrate the functions:
`1/(x + x log x)`
Integrate the functions:
`(e^(2x) - 1)/(e^(2x) + 1)`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following function w.r.t. x:
`(10x^9 +10^x.log10)/(10^x + x^10)`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int 1/(4x^2 - 20x + 17)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
`int ("x + 2")/(2"x"^2 + 6"x" + 5)"dx" = "p" int (4"x" + 6)/(2"x"^2 + 6"x" + 5) "dx" + 1/2 int "dx"/(2"x"^2 + 6"x" + 5)`, then p = ?
Choose the correct alternative from the following.
`int "dx"/(("x" - "x"^2))`=
`int (2(cos^2 x - sin^2 x))/(cos^2 x + sin^2 x)` dx = ______________
`int (log x)/(log ex)^2` dx = _________
`int (sin4x)/(cos 2x) "d"x`
`int ("e"^(2x) + "e"^(-2x))/("e"^x) "d"x`
`int sqrt(x) sec(x)^(3/2) tan(x)^(3/2)"d"x`
`int x/(x + 2) "d"x`
State whether the following statement is True or False:
`int3^(2x + 3) "d"x = (3^(2x + 3))/2 + "c"`
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
Evaluate `int(1+ x + x^2/(2!)) dx`
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate `int1/(x(x - 1))dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate `int (1)/(x(x - 1))dx`
Evaluate:
`int(sqrt(tanx) + sqrt(cotx))dx`
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate `int1/(x(x-1))dx`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
