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
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`
Integrate the function in `x^2e^x`.
Integrate the function in x log 2x.
Integrate the function in tan-1 x.
Integrate the function in ex (sinx + cosx).
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 e2x sin x.
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
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 tan(sin^-1 x)*dx` =
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Integrate the following w.r.t.x : cot–1 (1 – x + x2)
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` 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 (sinx)/(1 + sin x) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int"e"^(4x - 3) "d"x` = ______ + c
`int 1/sqrt(x^2 - 8x - 20) "d"x`
∫ log x · (log x + 2) dx = ?
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
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`
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Find: `int e^x.sin2xdx`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
`int1/(x+sqrt(x)) dx` = ______
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
Evaluate:
`int e^(logcosx)dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate:
`int x^2 cos x dx`
Integration by parts is a method of integration based on which rule of differentiation?
For \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx,\] which functions are chosen as the first function and the second function?
Which substitution can also be used before integrating by parts for \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx?\]
Let \[I=\int e^x\sin x\,dx.\] After applying integration by parts once, which equation is obtained?
Evaluate \[\int e^x\sin x\,dx.\]
Repeated parts may be needed for which pair of integrals?
