Advertisements
Advertisements
प्रश्न
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Advertisements
उत्तर
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
संबंधित प्रश्न
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
`x/(e^(x^2))`
Integrate the functions:
`sin x/(1+ cos x)`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Write a value of
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of
Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Choose the correct options from the given alternatives :
`int (cos2x - 1)/(cos2x + 1)*dx` =
Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Evaluate `int (5"x" + 1)^(4/9)` dx
`int 1/(cos x - sin x)` dx = _______________
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int ("e"^x(x - 1))/(x^2) "d"x` = ______
`int sqrt(1 + sin2x) dx`
`int logx/x "d"x`
`int(sin2x)/(5sin^2x+3cos^2x) dx=` ______.
`int sec^6 x tan x "d"x` = ______.
`int 1/(sinx.cos^2x)dx` = ______.
The value of `sqrt(2) int (sinx dx)/(sin(x - π/4))` is ______.
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
Evaluated the following
`int x^3/ sqrt (1 + x^4 )dx`
Evaluate `int(1 + x + x^2/(2!))dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate:
`int sin^3x cos^3x dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
