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
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in x sin 3x.
Integrate the function in x sec2 x.
Integrate the function in tan-1 x.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Integrate the function in `sin^(-1) ((2x)/(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 x.sin^-1 x.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int cos(root(3)(x)).dx`
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 (x- sinx)/(1 - cosx)*dx` =
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^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: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Evaluate:
∫ (log x)2 dx
`int 1/sqrt(2x^2 - 5) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int log x * [log ("e"x)]^-2` dx = ?
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
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.
Solve: `int sqrt(4x^2 + 5)dx`
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
`int(1-x)^-2 dx` = ______
`int1/sqrt(x^2 - a^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:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)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)
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
