Advertisements
Advertisements
Question
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
Advertisements
Solution
Let I = `int "dx"/sqrt(4"x"^2 - 5)`
`= int 1/(sqrt (4("x"^2 - 5/4)))`dx
`= 1/2 int 1/(sqrt("x"^2 - ((sqrt5)/2)^2))` dx
`= 1/2 log |"x" + sqrt("x"^2 - (sqrt5/2)^2)|` + c
∴ I = `1/2 log |"x" + sqrt("x"^2 - 5/4)|` + c
APPEARS IN
RELATED QUESTIONS
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in `x^2e^x`.
Integrate the function in x sec2 x.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following:
`int x tan^-1 x . dx`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Evaluate the following.
∫ x log x dx
`int (sinx)/(1 + sin x) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int logx/(1 + logx)^2 "d"x`
`int 1/sqrt(x^2 - 9) dx` = ______.
`int(logx)^2dx` equals ______.
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^2) (x^5 + 2x^3)dx`.
`intsqrt(1+x) 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
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`intcos^-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`
Evaluate `int tan^-1x dx`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate the following.
`intx^3e^(x^2) dx`
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
The value of `int (x sin^-1)/(sqrt(1 - x^2)) dx` is equal to:
`∫ sin^(−1)` xdx is equal to ______.
