Advertisements
Advertisements
Question
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Advertisements
Solution 1
`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`.
Solution 2
`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
RELATED QUESTIONS
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Integrate the functions:
`(sin^(-1) x)/(sqrt(1-x^2))`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`1/(1 - tan x)`
Write a value of\[\int \log_e x\ dx\].
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Integrate the following w.r.t. x : x3 + x2 – x + 1
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Integrate the following functions w.r.t. x : `(1)/(4x + 5x^-11)`
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Integrate the following functions w.r.t. x : cos7x
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Evaluate the following : `int (1)/(x^2 + 8x + 12).dx`
Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Evaluate the following : `int (logx)2.dx`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` dx
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate `int 1/((2"x" + 3))` dx
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int (2 + cot x - "cosec"^2x) "e"^x "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 sec^6 x tan x "d"x` = ______.
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
`int cos^3x 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)
Evaluate `int1/(x(x - 1))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)
Evaluate.
`int (5x^2-6x+3)/(2x-3)dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
