Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`
Advertisements
Solution
Let I = `int(x^n - 1)/sqrt(1 + 4x^n).dx`
Put xn = t
∴ nxn–1 dx = dt
∴ xn–1 dx = `dt/n`
∴ I = `int (1)/sqrt(1 + 4t).dt/n`
= `(1)/nint(1 + 4t)^(-1/2)dt`
= `1/n.((1 + 4t)^(1/2))/(1/2) xx (1)/(4) + c`
= `(1)/(2n).sqrt(1 + 4x^n) + c`.
APPEARS IN
RELATED QUESTIONS
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Evaluate : `∫1/(3+2sinx+cosx)dx`
Write a value of
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
`int "dx"/(9"x"^2 + 1)= ______. `
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integrals : `int cos^2x.dx`
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`
Integrate the following functions w.r.t. x : `(1)/(4x + 5x^-11)`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
State whether the following statement is True or False.
If ∫ x f(x) dx = `("f"("x"))/2`, then find f(x) = `"e"^("x"^2)`
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int logx/x "d"x`
`int (cos2x)/(sin^2x) "d"x`
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate the following.
`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`
Evaluate `int (1)/(x(x - 1))dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate the following.
`int x^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).
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate `int(1 + x + x^2 / (2!))dx`
Evaluate `int 1/(x(x-1)) dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
For \[\sqrt{\frac{x-\alpha}{\beta-x}}\] or \[\sqrt{(x-\alpha)(\beta-x)}\], where \[\beta>\alpha\], which substitution is used?
When applying substitution, what must always be rewritten in terms of the new variable?
