Advertisements
Advertisements
प्रश्न
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Advertisements
उत्तर १
`int(4x + 3)/(2x + 1).dx`
= `int((2(2x + 1) + 1))/(2x + 1).dx`
= `int ((2(2x + 1))/(2x + 1) + 1/(2x + 1)).dx`
= `2 int 1 dx + int 1/(2x + 1).dx`
= `2x + (1)/(2) log|2x + 1| + c`.
उत्तर २
`int(4x + 3)/(2x + 1).dx`
`u = 2x + 1=> (du)/(dx) = 2 => dx = (du)/2`
Now express the numerator 4x + 3 in terms of u:
`x = (u-1)/2`
`4x+3=4xx (u-1)/2 +3 = 2(u-1)+3=2u-2+3=2u+1`
`int(4x+3)/(2x+1) dx = int(2u+1)/uxx(du)/2`
`= 1/2int(2+1/u)du`
`1/2 int (2+1/u) du=1/2(2u+ln|u|)+C=u+1/2 ln|u|+C`
`=(2x+1)+1/2ln |2x+1|+C`
APPEARS IN
संबंधित प्रश्न
Evaluate :`intxlogxdx`
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Integrate the functions:
`(log x)^2/x`
Integrate the functions:
`1/(x + x log x)`
Write a value of
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`
Evaluate the following integrals : `int sqrt((9 - x)/x).dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Evaluate `int (1 + x + x^2/(2!))`dx
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` dx
Evaluate `int (5"x" + 1)^(4/9)` dx
Evaluate `int "x - 1"/sqrt("x + 4")` dx
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
`int (cos2x)/(sin^2x) "d"x`
`int x^3"e"^(x^2) "d"x`
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
`int(log(logx) + 1/(logx)^2)dx` = ______.
If `int sinx/(sin^3x + cos^3x)dx = α log_e |1 + tan x| + β log_e |1 - tan x + tan^2x| + γ tan^-1 ((2tanx - 1)/sqrt(3)) + C`, when C is constant of integration, then the value of 18(α + β + γ2) is ______.
`int (logx)^2/x dx` = ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate `int (1+x+x^2/(2!))dx`
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4)) dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate `int(1 + x + x^2 / (2!))dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
