Advertisements
Advertisements
प्रश्न
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
विकल्प
`x cot (x/2) + c`
`- x cot (x/2) + c`
`cot (x/2) + c`
`x tan (x/2) + c`
Advertisements
उत्तर
`- x cot (x/2) + c`
[ Hint : `int (x- sinx)/(1 - cosx)*dx = int (x - 2sin(x/2)cos(x/2))/(2sin^2 (x/2))*dx`
= `(1)/(2) int x"cosec"^2(x/2)*dx - int cot(x/2)*dx`
= `(1)/(2) [x int "cosec"^2 (x/2)*dx - int [d/dx(x) int "cosec"^2(x/2)^(dx)]*dx - int cot(x/2)*dx`
= `(1)/(2)[x{(-cot(x/2))/((1/2))} - int1* (-cot(x/2))/((1/2))*dx - intcot(x/2)*dx`
= `xcot(x/2) + int cot(x/2)*dx - int cot(x/2)*dx`
= `- x cot(x/2) + c`].
APPEARS IN
संबंधित प्रश्न
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Integrate the function in x sin x.
Integrate the function in `x^2e^x`.
Integrate the function in x tan-1 x.
Integrate the function in x cos-1 x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in (x2 + 1) log x.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following: `int x.sin^-1 x.dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Choose the correct options from the given alternatives :
`int (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
`int (sinx)/(1 + sin x) "d"x`
`int 1/(4x + 5x^(-11)) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int sin4x cos3x "d"x`
`int"e"^(4x - 3) "d"x` = ______ + c
Evaluate `int 1/(x(x - 1)) "d"x`
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int(1-x)^-2 dx` = ______
`int1/sqrt(x^2 - a^2) dx` = ______
`intsqrt(1+x) dx` = ______
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate:
`int e^(logcosx)dx`
Complete the following activity:
`int_0^2 dx/(4 + x - x^2) `
= `int_0^2 dx/(-x^2 + square + square)`
= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`
= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`
= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
The value of `inta^x.e^x dx` equals
`∫ sin^(−1)` xdx is equal to ______.
Which expression is the integration-by-parts form for a product \(f(x)g(x)\)?
For \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx,\] which functions are chosen as the first function and the second function?
