Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Advertisements
Solution
Let I = `int x^5sqrt(a^2 + x^2).dx`
Put, a2 + x2 = t
∴ 2x dx = dt
∴ x dx = `(1)/(2)dt`
Also, x2 = t – a2
I = `int x^2. x^2sqrt(a^2 + x^2)x dx`
=` int(t - a^2)^2 sqrt(t). dt`
= `(1)/(2) int (t^2 - 2a^2t + a^4)sqrt(t). dt`
= `(1)/(2) int (t^(5/2) - 2a^2t^(3/2) + a^4t^(1/2))dt`
= `(1)/(2) int t^(5/2) dt - a^2 int t^(3/2) dt + a^4/2 int t^(1/2) dt`
= `(1)/(2). (t^(7/2))/((7/2)) - a^2. (t^(5/2))/((5/2)) + a^4/2.(t^(3/2))/((3/2) )+ c`
= `(1)/(7)(a^2 + x^2)^(7/2) - (2a^2)/(5)(a^2 + x^2)^(5/2) + a^4/(3)(a^2 + x^2)^(3/2) + c`
APPEARS IN
RELATED QUESTIONS
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
Integrate the functions:
`cos sqrt(x)/sqrtx`
Integrate the functions:
`cos x /(sqrt(1+sinx))`
Integrate the functions:
`sin x/(1+ cos x)`
Write a value of
Write a value of
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
If f'(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`
`int(log(logx))/x "d"x`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
`int(5x + 2)/(3x - 4) dx` = ______
`int(7x - 2)^2dx = (7x -2)^3/21 + c`
`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
Evaluate the following.
`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`
if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`
Evaluate `int(1 + x + x^2/(2!))dx`
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3) dx`
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
Evaluate.
`int (5x^2 -6x + 3)/(2x -3)dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
