Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t.x : log (x2 + 1)
उत्तर
Let I = `int log (x^2 + 1)*dx`
= `int [log (x^2 + 1)]*1dx`
= `[log(x^2 + 1)] int 1dx - int [d/dx{log (x^2 + 1)} int 1dx]*dx`
= `[log (x^2 + 1)]*x - int 1/(x^2 + 1)*dx (x^2 + 1)*xdx`
= `xlog(x^2 + 1) - int (2x^2)/(x^2 + 1)*dx`
= `xlog (x^2 + 1) - int (2x^2 + 2 - 2)/(x^2 + 1)*dx`
= `xlog(x^2 + 1) - int[(2(x^2 + 1))/(x^2 + 1) - 2/(x^2 + 1)]*dx`
= `xlog(x^2 + 1) - int[2 int 1dx - 2 int 1/(x^2 + 1)*dx]`
= x log (x2 + 1) – 2x + 2 tan–1 x + c.
APPEARS IN
संबंधित प्रश्न
Integrate the function in x log x.
Integrate the function in x log 2x.
Integrate the function in x tan-1 x.
Integrate the function in x cos-1 x.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x^2 sin 3x dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int cos sqrt(x).dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following : `int x.cos^3x.dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `sqrt(x^2 + 2x + 5)`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
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` =
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 sin (log x)*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int "e"^"x" (1/"x" - 1/"x"^2)`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int sin4x cos3x "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int ("d"x)/(x - x^2)` = ______
`int 1/x "d"x` = ______ + c
Evaluate `int 1/(x log x) "d"x`
`int "e"^x x/(x + 1)^2 "d"x`
`int logx/(1 + logx)^2 "d"x`
`int cot "x".log [log (sin "x")] "dx"` = ____________.
`int log x * [log ("e"x)]^-2` dx = ?
Find `int_0^1 x(tan^-1x) "d"x`
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
`int 1/sqrt(x^2 - a^2)dx` = ______.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
`int(1-x)^-2 dx` = ______
`intsqrt(1+x) dx` = ______
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
`int1/(x+sqrt(x)) dx` = ______
`inte^(xloga).e^x dx` is ______
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate:
`inte^x sinx dx`
Evaluate:
`int (logx)^2 dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate the following.
`intx^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).
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`
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
Evaluate `int(1 + x + x^2/(2!))dx`.
