Advertisements
Advertisements
प्रश्न
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
Advertisements
उत्तर
Let I = `int (2x + 1)/((x + 1)(x - 2)) "d"x`
Let `(2x + 1)/((x + 1)(x - 2)) = "A"/(x + 1) + "B"/(x - 2)`
∴ 2x + 1 = A(x – 2) + B(x + 1) ......(i)
Putting x = – 1 in (i), we get
2(– 1) + 1 = A(– 1 – 2) + B(0)
∴ – 1 = – 3A
∴ A = `1/3`
Putting x = 2 in (i), we get
2(2) + 1 = A(0) + B(2 + 1)
∴ 5 = 3B
∴ B = `5/3`
∴ `(2x + 1)/((x + 1)(x - 2)) = ((1/3))/(x + 1) + ((5/3))/(x - 2)`
∴ I = `int(((1/3))/(x + 1) + ((5/3))/(x - 2)) "d"x`
= `1/3 int 1/(x + 1) "d"x + 5/3 int 1/(x - 2) "d"x`
∴ I = `1/3 log|x + 1| + 5/3 log|x - 2| + "c"`
संबंधित प्रश्न
Integrate the function in x (log x)2.
Integrate the function in `e^x (1/x - 1/x^2)`.
Evaluate the following : `int sin θ.log (cos θ).dθ`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Integrate the following w.r.t.x : log (log x)+(log x)–2
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int 1/x "d"x` = ______ + c
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
∫ log x · (log x + 2) dx = ?
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
`int 1/sqrt(x^2 - a^2)dx` = ______.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
Solution of the equation `xdy/dx=y log y` is ______
`int1/(x+sqrt(x)) dx` = ______
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
For \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx,\] which functions are chosen as the first function and the second function?
Evaluate \[\int x\cos x\,dx.\]
Evaluate \[\int e^x\sin x\,dx.\]
