Advertisements
Advertisements
प्रश्न
Evaluate the following : `int cos sqrt(x).dx`
Advertisements
उत्तर
Let I = `int cos sqrt(x).dx`
Put `sqrt(x) = t`
∴ x = t2
∴ dx = 2t .dt
∴ I = `int(cost)2t.dt`
= `int 2t cos t.dt`
= `2t int cos.dt - int [d/dt (2t) int cos t.dt ].dt`
= `2tsint - int 2 sint.dt`
= 2t sin t + 2 cos t + c
= `2[sqrt(x)sinsqrt(x) + cos sqrt(x)] + c`.
APPEARS IN
संबंधित प्रश्न
Integrate the function in x sin x.
Integrate the function in x2 log x.
Integrate the function in `(xe^x)/(1+x)^2`.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
`int e^x sec x (1 + tan x) dx` equals:
Prove that:
`int sqrt(x^2 + a^2)dx = x/2 sqrt(x^2 + a^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`
Evaluate the following : `int x^2.log x.dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Choose the correct alternative from the following.
`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` =
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Evaluate `int 1/(x log x) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
`int cot "x".log [log (sin "x")] "dx"` = ____________.
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
`int tan^-1 sqrt(x) "d"x` is equal to ______.
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Solve: `int sqrt(4x^2 + 5)dx`
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`int_0^1 x tan^-1 x dx` = ______.
Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int(1-x)^-2 dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
For the special integral \[\int e^x\left[f(x)+f'(x)\right]dx,\] what is the antiderivative?
Which differentiation identity verifies the special integral \[\int e^x\left[f(x)+f'(x)\right]dx=e^xf(x)+C?\]
Repeated parts may be needed for which pair of integrals?
