Advertisements
Advertisements
Question
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Advertisements
Solution
Let I = `int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Put f(x) = (log x)2
∴ f '(x) = `(2 log "x")/"x"`
∴ I = ∫ ex [f(x) + f '(x)] + dx
= ex f(x) + c
∴ I = ex (log x)2 + c
Notes
The answer in the textbook is incorrect.
APPEARS IN
RELATED QUESTIONS
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 log x.
Integrate the function in x log 2x.
Integrate the function in x sin−1 x.
Integrate the function in `(xe^x)/(1+x)^2`.
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
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 (1)/(cosx - cos^2x)*dx` =
Choose the correct options from the given alternatives :
`int [sin (log x) + cos (log x)]*dx` =
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int "x"^2 *"e"^"3x"`dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int(x + 1/x)^3 dx` = ______.
Find `int_0^1 x(tan^-1x) "d"x`
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
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`
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 ______.
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate `int tan^-1x 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^3e^(x^2)dx`
The value of `inta^x.e^x dx` equals
Evaluate `int(1 + x + x^2/(2!))dx`.
