Advertisements
Advertisements
Question
Integrate the following w.r.t.x : log (x2 + 1)
Advertisements
Solution
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
RELATED QUESTIONS
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 log x.
Integrate the function in x cos-1 x.
Integrate the function in x sec2 x.
Integrate the function in tan-1 x.
Integrate the function in `(xe^x)/(1+x)^2`.
Integrate the function in `e^x (1/x - 1/x^2)`.
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following : `int cos sqrt(x).dx`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` 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(x + 1/x)^3 dx` = ______.
Evaluate `int 1/(x log x) "d"x`
`int cot "x".log [log (sin "x")] "dx"` = ____________.
`int log x * [log ("e"x)]^-2` dx = ?
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
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.
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 ______.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
Solution of the equation `xdy/dx=y log y` is ______
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))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`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
The value of `int (x sin^-1)/(sqrt(1 - x^2)) dx` is equal to:
`∫ sin^(−1)` xdx is equal to ______.
Integration by parts is a method of integration based on which rule of differentiation?
Which pair is listed as Trigonometric in the LIATE rule?
Which substitution can also be used before integrating by parts for \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx?\]
When applying integration by parts to \(\log x\) and inverse trig, what should they be multiplied by?
