Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t.x : log (x2 + 1)
Advertisements
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 : sec3 x w. r. t. x.
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
Integrate the function in x log x.
Integrate the function in x log 2x.
Integrate the function in `(xe^x)/(1+x)^2`.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following : `int cos sqrt(x).dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
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: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Evaluate:
∫ (log x)2 dx
`int (sinx)/(1 + sin x) "d"x`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int(x + 1/x)^3 dx` = ______.
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
`int "e"^x x/(x + 1)^2 "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Find: `int e^x.sin2xdx`
`int(logx)^2dx` equals ______.
Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
Evaluate:
`int e^(logcosx)dx`
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`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
`∫ sin^(−1)` xdx is equal to ______.
Evaluate \[\int e^x\sin x\,dx.\]
When applying integration by parts to \(\log x\) and inverse trig, what should they be multiplied by?
