Advertisements
Advertisements
प्रश्न
Evaluate the following : `int x^2tan^-1x.dx`
Advertisements
उत्तर
Let I = `int x^2 tan^-1 x.dx`
= `int(tan^-1x).x^2dx`
= `(tan^-1x) int x^2.dx - int[{d/dx(tan^-1x) int x^2.dx}].dx`
= `(tan^-1 x)(x^3/3) - int (1/(1 + x^2))(x^3/3).dx`
= `x3/(3) tan^-1x - (1)/(3) (x(x^2 + 1) - x)/(x^2 + 1).dx`
= `x^3/(3) tan^-1x - (1)/(3)[int{x - x/(x^2 + 1)}.dx]`
= `x^3/(3) tan^-1x - (1)/(3)[int x.dx - (1)/(2) int(2x)/(x^2 + 1).dx]`
= `x^3/(3)tan^-1x - (1)/(3) [x^2/(2) - (1)/(2)log|x^2 + 1|] + c`
...`[because d/dx(x^2 + 1) = 2x and int (f'(x))/f(x) dx = log|f(x)| + c]`
= `x^3/(3)tan^-1x - x^2/(6) + (1)/(6) log|x^2 + 1| + 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 x sin x.
Integrate the function in x log 2x.
Integrate the function in x2 log x.
Integrate the function in x tan-1 x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int cos sqrt(x).dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
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 (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =
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` =
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
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
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
Evaluate:
∫ (log x)2 dx
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int sin4x cos3x "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
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
∫ log x · (log x + 2) dx = ?
Evaluate the following:
`int_0^pi x log sin x "d"x`
If u and v ore differentiable functions of x. then prove that:
`int uv dx = u intv dx - int [(du)/(d) intv dx]dx`
Hence evaluate `intlog x dx`
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` = ______.
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^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
Evaluate the following.
`int x^3 e^(x^2) 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
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`inte^x sinx dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int (1 + x + x^2/(2!))dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
