Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Advertisements
उत्तर
Let I = `int log (1 + x)^((1 + x)).dx`
= `int (1 + x)log(1 + x).dx`
= `int [log(1 + x)] (1 + x).dx`
= `[log(1 + x) int (1 + x).dx - int[d/dt {log(1 + x)} int (1 + x).dx].dx`
= `[log (1 + x)] [(1 + x)^2/2] - int 1/(x + 1).(x + 1)^2/(2).dx`
= `(x + 1)^2/(2).log(1 + x) - (1)/(2) int (x + 1).dx`
= `(x + 1)^2/(2).log (1 + x) - (1)/(2).(x + 1)^2/(2) + c`
= `(x + 1)^2/(2)[log (1 + x) - 1/2] + c`.
APPEARS IN
संबंधित प्रश्न
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 x2 log x.
Integrate the function in ex (sinx + cosx).
Find :
`∫(log x)^2 dx`
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int e^(2x).cos 3x.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(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
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)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int 1/x "d"x` = ______ + c
`int "e"^x x/(x + 1)^2 "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Find `int_0^1 x(tan^-1x) "d"x`
Evaluate the following:
`int_0^pi x log sin x "d"x`
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
Find: `int e^x.sin2xdx`
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int1/(x+sqrt(x)) dx` = ______
Solve the differential equation (x2 + y2) dx - 2xy dy = 0 by completing the following activity.
Solution: (x2 + y2) dx - 2xy dy = 0
∴ `dy/dx=(x^2+y^2)/(2xy)` ...(1)
Puty = vx
∴ `dy/dx=square`
∴ equation (1) becomes
`x(dv)/dx = square`
∴ `square dv = dx/x`
On integrating, we get
`int(2v)/(1-v^2) dv =intdx/x`
∴ `-log|1-v^2|=log|x|+c_1`
∴ `log|x| + log|1-v^2|=logc ...["where" - c_1 = log c]`
∴ x(1 - v2) = c
By putting the value of v, the general solution of the D.E. is `square`= cx
`inte^(xloga).e^x dx` is ______
`int logx dx = x(1+logx)+c`
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate:
`inte^x sinx dx`
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Under the LIATE rule, which type of function has priority \(L\)?
Which function is an example under priority \(I\) in the LIATE rule?
Evaluate \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx.\]
Let \[I=\int e^x\sin x\,dx.\] After applying integration by parts once, which equation is obtained?
Which differentiation identity verifies the special integral \[\int e^x\left[f(x)+f'(x)\right]dx=e^xf(x)+C?\]
Repeated parts may be needed for which pair of integrals?
