Advertisements
Advertisements
प्रश्न
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
Advertisements
उत्तर
Let I = `int (2x + 3)/(2x^2 + 3x - 1).dx`
Let 2x + 3 = `"A"[d/dx(2x^2 + 3x - 1)] + "B"`
= A(4x + 3) + B
∴ 2x + 3 = 4Ax + (3A + B)
Comapring the coefficientof x and constant on both sides, we get
4A = 2 and 3A + B = 3
∴ A = `(1)/(2) and 3(1/2) + "B"` = 3
∴ B = `(3)/(2)`
∴ 2x + 3 = `(1)/(2)(4x + 3) + (3)/(2)`
∴ I = `int (1/2(4x + 3) + (3)/(2))/(2x^2 + 3x - 1).dx`
= `(1)/(2) int (4x + 3)/(2x^2 + 3x - 1).dx + (3)/(2) int (1)/(2x^2 + 3x - 1).dx`
= `(1)/(2)"I"_1 + (3)/(2)"I"_2`
I1 is of the type `int (f'(x))/f(x)dx = log|f(x)| + c`
∴ I1 = log |2x2 + 3x – 1| + c1
I2 = `int (1)/(2x^2 + 3x - 1).dx`
= `(1)/(2) int (1)/(x^2 + 3/2x - 1/2).dx`
= `(1)/(2) int (1)/((x^2 + 3/2x + 9/16) - 9/16 - 1/2).dx`
= `(1)/(2) int (1)/((x + 3/4)^2 - (sqrt(17)/4)^2).dx`
= `(1)/(2) xx (1)/(2 xx sqrt(17)/(4))log|(x + 3/4 - sqrt(17)/4)/(x + 3/4 + sqrt(17)/4)| + c_2`
= `(1)/sqrt(17)log|(4x + 3 - sqrt(17))/(4x + 3 + sqrt(17))| + c_2`
∴ I = `(1)/(2)log|2x^2 + 3x - 1| + (3)/(2sqrt(17))log|(4x + 3 - sqrt(17))/(4x + 3 + sqrt(17))| + c`, where c = c + c2.
APPEARS IN
संबंधित प्रश्न
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`
Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following:
`int sinx/(sin 3x) dx`
Evaluate the following.
`int 1/(4x^2 - 20x + 17)` dx
To find the value of `int ((1 + log x) )/x dx` the proper substitution is ______.
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
Evaluate: `int log ("x"^2 + "x")` dx
`int 1/(cos x - sin x)` dx = _______________
`int 1/(xsin^2(logx)) "d"x`
`int (cos2x)/(sin^2x) "d"x`
`int x^3"e"^(x^2) "d"x`
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
Evaluate `int (1+x+x^2/(2!))dx`
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
Evaluate `int(1+ x + x^2/(2!)) dx`
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
`int dx/((x+2)(x^2 + 1))` ...(given)
`1/(x^2 +1) dx = tan ^-1 + c`
Evaluate `int (1)/(x(x - 1))dx`
Evaluate the following.
`int x^3/sqrt(1+x^4) dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4)) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Evaluate `int1/(x(x-1))dx`
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate `int 1/(x(x-1)) dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
For \[\int\frac{\sin x}{\sin(x+a)}\,dx\], which substitution gives \[dx=dt\]?
For indefinite integrals, what should be done after integration in the new variable?
