Advertisements
Advertisements
Question
Evaluate:
`int1/(x^2 + 25)dx`
Advertisements
Solution
Let I = `int1/(x^2 + 25)dx`
= `1/(x^2 + (5)^2)dx`
= `1/5 tan^-1 x/5 + c`
APPEARS IN
RELATED QUESTIONS
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Integrate the function in x sin 3x.
Integrate the function in x2 log x.
Integrate the function in x cos-1 x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
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/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
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)/(xsin^2(logx)`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : e2x sin x cos x
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate:
∫ (log x)2 dx
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int cot "x".log [log (sin "x")] "dx"` = ____________.
Find `int_0^1 x(tan^-1x) "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.
Find: `int e^x.sin2xdx`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) 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 ______.
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`int (logx)^2 dx`
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`
Evaluate the following.
`intx^3e^(x^2) dx`
Evaluate `int (1 + x + x^2/(2!))dx`
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
