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.
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 sin x.
Integrate the function in `x^2e^x`.
Integrate the function in x log 2x.
Integrate the function in x cos-1 x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in tan-1 x.
Integrate the function in ex (sinx + cosx).
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: `int logx/x.dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
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 sin (log x)*dx` =
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
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 : log (log x)+(log x)–2
Integrate the following w.r.t.x : sec4x cosec2x
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int "dx"/("9x"^2 - 25)`
`int (sinx)/(1 + sin x) "d"x`
`int 1/sqrt(2x^2 - 5) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int 1/x "d"x` = ______ + c
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
Find `int_0^1 x(tan^-1x) "d"x`
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
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 e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`inte^(xloga).e^x dx` is ______
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`int x sqrt(1 + x^2) dx`
