Advertisements
Advertisements
Question
Integrate the functions:
`xsqrt(x + 2)`
Advertisements
Solution
Let `I = int x sqrt(x + 2)` dx
Taking `sqrt(x + 2) = t`
or x + 2 = t2
or dx = 2t dt
Hence, `I = int (t^2 - 2). t. 2t dt`
`= int 2t^4 - 4t^2 dt`
`= 2 int (t^4 - 2t^2)` dt
`= 2 int t^4 dt - 4 int` t2 dt
`= 2/5 t^5 - 4/3 t^3 + C`
`=> 2/5 (x + 2)^(5/2) - 4/3 (x + 2)^(3/2) + C`
APPEARS IN
RELATED QUESTIONS
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Write a value of
Write a value of
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Evaluate `int (1 + x + x^2/(2!))`dx
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate: `int 1/(sqrt("x") + "x")` dx
Evaluate: `int "e"^sqrt"x"` dx
`int sqrt(x) sec(x)^(3/2) tan(x)^(3/2)"d"x`
`int(log(logx))/x "d"x`
`int1/(4 + 3cos^2x)dx` = ______
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.
`int sqrt(x^2 - a^2)/x dx` = ______.
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate:
`int sqrt((a - x)/x) dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate `int 1/(x(x-1))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).
Which standard substitution is used for \[\sqrt{\mathrm{a}^2-x^2}\], \[\frac{1}{\sqrt{\mathrm{a}^2-x^2}}\], or \[\mathrm{a}^2-x^2\]?
Which substitution is appropriate for \[\sqrt{\frac{x}{x-a}}\], \[\sqrt{\frac{x-a}{x}}\], \[\sqrt{x(x-\mathrm{a})}\], or \[\frac{1}{\sqrt{x(x-\mathrm{a})}}\]?
After choosing \[u=g(x)\], what is the next step?
