Advertisements
Advertisements
प्रश्न
Evaluate the following : `int log(logx)/x.dx`
Advertisements
उत्तर
Let I = `int log(logx)/x.dx`
= `int log(logx). 1/xdx`
Put log = t
∴ `1/x.dx = dt`
∴ I = `int logt dt`
= `int (logt).1dt`
= `(logt) int 1dt - int[d/d (logt) int 1dt]dt`
= `(log t)t - int 1/t xx tdt`
= `t log t - int 1dt`
= t logt t – t + c
= t(log t – 1) + c
= (log x).[log(log x) – 1] + c.
APPEARS IN
संबंधित प्रश्न
Integrate the function in x cos-1 x.
Integrate the function in `e^x (1/x - 1/x^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 cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
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 : cosec (log x)[1 – cot (log 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 sin (log x)*dx` =
Choose the correct options from the given alternatives :
`int [sin (log x) + cos (log x)]*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int 1/sqrt(2x^2 - 5) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int(x + 1/x)^3 dx` = ______.
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
Evaluate `int 1/(x log x) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
`int 1/sqrt(x^2 - 9) dx` = ______.
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
Find: `int e^x.sin2xdx`
`int(logx)^2dx` equals ______.
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 ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int1/(x+sqrt(x)) dx` = ______
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Evaluate the following:
`intx^3e^(x^2)dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
The value of `inta^x.e^x dx` equals
