Advertisements
Advertisements
प्रश्न
Evaluate the following : `int cos sqrt(x).dx`
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
Prove that:
`int sqrt(a^2 - x^2) dx = x/2 sqrt(a^2 - x^2) + a^2/2sin^-1(x/a)+c`
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
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
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
Integrate the function in x sin 3x.
Integrate the function in x (log x)2.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
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` =
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : log (x2 + 1)
Evaluate the following.
∫ x log x dx
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int 1/(4x + 5x^(-11)) "d"x`
`int 1/sqrt(2x^2 - 5) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int 1/x "d"x` = ______ + c
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int"e"^(4x - 3) "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Evaluate `int 1/(x log x) "d"x`
`int "e"^x x/(x + 1)^2 "d"x`
`int logx/(1 + logx)^2 "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
∫ log x · (log x + 2) dx = ?
`int log x * [log ("e"x)]^-2` dx = ?
Find `int_0^1 x(tan^-1x) "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
`int tan^-1 sqrt(x) "d"x` is equal to ______.
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
`int 1/sqrt(x^2 - 9) dx` = ______.
`int 1/sqrt(x^2 - a^2)dx` = ______.
If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.
If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int1/sqrt(x^2 - a^2) dx` = ______
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
`int(xe^x)/((1+x)^2) dx` = ______
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3 e^(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.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
