Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Advertisements
उत्तर
Let I = `int e^x/x [x (logx)^2 + 2log x].dx`
= `int e^x [(logx)^2 + (2logx)/x].dx`
Put f(x) = (log x)2
∴ f'(x) = `d/dx (logx)^2`
= `2 (logx).d/dx (logx)`
= `(2logx)/x`
∴ I = `int e^x [f(x) + f'(x)].dx`
= ex . f(x) + c
= ex . (log x)2 + c.
APPEARS IN
संबंधित प्रश्न
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−1 x.
Integrate the function in (sin-1x)2.
Integrate the function in `(xe^x)/(1+x)^2`.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in `e^x (1/x - 1/x^2)`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Integrate the function in e2x sin x.
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
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` = ______.
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : cot–1 (1 – x + x2)
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : log (log x)+(log x)–2
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
`int 1/(4x + 5x^(-11)) "d"x`
Evaluate `int 1/(x(x - 1)) "d"x`
`int "e"^x x/(x + 1)^2 "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
∫ log x · (log x + 2) dx = ?
`int cot "x".log [log (sin "x")] "dx"` = ____________.
`int "e"^x [x (log x)^2 + 2 log x] "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 x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`inte^x sinx dx`
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)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 e^(x^2)dx`
Repeated parts may be needed for which pair of integrals?
