Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Advertisements
Solution
Let I = `ff(2x + 1)sqrt(x + 2).dx`
Put x + 2 = t
∴ dx = dt
Also, x = t – 2
∴ 2x + 1 = 2(t – 2) + 1 = 2t – 3
∴ I = `int (2t - 3)sqrt(t) dt`
= `int (2t^(3/2) - 3t^(1/2))dt`
= `2int t^(3/2)dt - 3 intt^(1/2)dt`
= `2.(t^5/2)/((5/2)) - 3 . (t^(3/2))/((3/2)) + c`
= `(4)/(5)(x + 2)^(5/2) - 2(x + 2)^(3/2) + c`.
APPEARS IN
RELATED QUESTIONS
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Integrate the functions:
`(1+ log x)^2/x`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Evaluate: `int 1/(x(x-1)) dx`
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `int sin x/cos^2x dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Integrate the following functions w.r.t. x : tan5x
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int 1/(sqrt"x" + "x")` dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
`int (2(cos^2 x - sin^2 x))/(cos^2 x + sin^2 x)` dx = ______________
`int (7x + 9)^13 "d"x` ______ + c
`int sec^6 x tan x "d"x` = ______.
If f'(x) = `x + 1/x`, then f(x) is ______.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
Which standard substitution is used for \[\sqrt{\mathrm{a}^2-x^2}\], \[\frac{1}{\sqrt{\mathrm{a}^2-x^2}}\], or \[\mathrm{a}^2-x^2\]?
For \[\sqrt{\frac{x-\alpha}{\beta-x}}\] or \[\sqrt{(x-\alpha)(\beta-x)}\], where \[\beta>\alpha\], which substitution is used?
What is \[\int\frac{\sin x}{\sin(x+a)}\,dx\]?
How should a substitution be chosen?
