Advertisements
Advertisements
Question
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Options
`x cot (x/2) + c`
`- x cot (x/2) + c`
`cot (x/2) + c`
`x tan (x/2) + c`
Solution
`- 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
RELATED QUESTIONS
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
Integrate the function in `x^2e^x`.
Integrate the function in tan-1 x.
Integrate the function in x (log x)2.
Integrate the function in (x2 + 1) log x.
Integrate the function in `(xe^x)/(1+x)^2`.
`int e^x sec x (1 + tan x) dx` equals:
Find :
`∫(log x)^2 dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
Evaluate: `int "dx"/(5 - 16"x"^2)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
Evaluate: ∫ (log x)2 dx
`int 1/(4x + 5x^(-11)) "d"x`
`int 1/sqrt(2x^2 - 5) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
Evaluate `int 1/(x log x) "d"x`
Evaluate `int 1/(4x^2 - 1) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int log x * [log ("e"x)]^-2` dx = ?
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
Find `int_0^1 x(tan^-1x) "d"x`
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
If u and v ore differentiable functions of x. then prove that:
`int uv dx = u intv dx - int [(du)/(d) intv dx]dx`
Hence evaluate `intlog x dx`
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int(logx)^2dx` equals ______.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
`intsqrt(1+x) dx` = ______
`inte^(xloga).e^x dx` is ______
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
Evaluate:
`int e^(logcosx)dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate the following.
`intx^3e^(x^2) 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)
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^3 e^(x^2)dx`
Evaluate `int(1 + x + x^2/(2!))dx`.
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
