Advertisements
Advertisements
प्रश्न
Evaluate the following : `int cos(root(3)(x)).dx`
Advertisements
उत्तर
Let I = `int cos(root(3)(x)).dx`
Put `root(3)(x)` = t
∴ x = t3
∴ dx = 3t2.dt
∴ I = `int 3t^2 cos t.dt`
= `3t^2 int cos t.dt - int [d/dt (3t)^2 int cos t.dt].dt`
= `3t^2 sint - int 6t sint.dt`
= `3t^2 sint - [6t sin t.dt - int {d/dt (6t) int sin t.dt }.dt]`
= `3t^2 sint - [6t (- cos t) - int 6( - cos t).dt]`
= 3t2 sin t + 6t cos t – 6 sin t + c
= 3(t2 – 2) sin t + 6t cos t + c
= `3(x^(2/3) - 2) sin(root(3)(x)) + 6root(3)(x) cos(root(3)(x)) + 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 (sin-1x)2.
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`
Find :
`∫(log x)^2 dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int cos sqrt(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^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Evaluate the following.
`int x^3 e^(x^2)`dx
`int ("x" + 1/"x")^3 "dx"` = ______
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int (sinx)/(1 + sin x) "d"x`
`int 1/(4x + 5x^(-11)) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int ("d"x)/(x - x^2)` = ______
`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 = ?
`int cot "x".log [log (sin "x")] "dx"` = ____________.
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
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`
`int 1/sqrt(x^2 - 9) dx` = ______.
`int_0^1 x tan^-1 x dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`int e^(logcosx)dx`
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.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int (1 + x + x^2/(2!))dx`
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^2e^(4x)dx`
