Advertisements
Advertisements
प्रश्न
Integrate the function `(x + 2)/sqrt(4x - x^2)`
Advertisements
उत्तर
Let `I = int (x + 2)/sqrt(4x - x^2) dx`
`= int (x + 2)/sqrt(- (x^2 - 4x + 4) + 4) dx`
`= int (x + 2)/sqrt(4 - (x - 2)^2) dx`
`= (x - 2 + 4)/sqrt(4 - (x - 2)^2) dx`
`= int (x - 2)/sqrt(4 - (x - 2)^2) dx + 4 int 1/sqrt(4 - (x - 2)^2) dx`
Let `I = I_1 + 4sin^-1 ((x - 2)/2) + C_1` .....(i)
Where `I_1 = int (x - 2)/ sqrt (4 - (x - 2)^2)dx`
Put 4 - (x - 2)2 = t
-2 (x - 2) dx = dt
`I_1 = 1/2 int dt/ sqrt ((2)^2 - t)`
`1/2 int dt/ sqrt (4 - t)`
`1/2 [(4 - t)^(-1/(2+1))/-(-1/2 + 1)] + C_2`
`= -sqrt ((4 - t)) + C_2`
`= - sqrt (4 - (x - 2)^2) + C_2`
`= - sqrt (4 - x^2 - 4 + 4x) + C_2`
`= - sqrt (4x - x^2) + C_2` ....(ii)
`I = - sqrt (4x - x^2) + 4sin^-1 ((x - 2)/2) + C` ... [from (i) and (ii)]
APPEARS IN
संबंधित प्रश्न
find : `int(3x+1)sqrt(4-3x-2x^2)dx`
Integrate the function `x^2/(1 - x^6)`
Integrate the function `(sec^2 x)/sqrt(tan^2 x + 4)`
Integrate the function `1/sqrt(7 - 6x - x^2)`
Integrate the function `1/sqrt(8+3x - x^2)`
Integrate the function `(x + 2)/sqrt(x^2 -1)`
Integrate the function `(6x + 7)/sqrt((x - 5)(x - 4))`
Integrate the function `(x+2)/sqrt(x^2 + 2x + 3)`
Integrate the function `(x + 3)/(x^2 - 2x - 5)`
Integrate the function `(5x + 3)/sqrt(x^2 + 4x + 10)`
`int dx/sqrt(9x - 4x^2)` equals:
Integrate the function:
`sqrt(4 - x^2)`
Integrate the function:
`sqrt(x^2 + 4x +1)`
Integrate the function:
`sqrt(x^2 + 4x - 5)`
Integrate the function:
`sqrt(1+ 3x - x^2)`
`int sqrt(1+ x^2) dx` is equal to ______.
`int sqrt(x^2 - 8x + 7) dx` is equal to ______.
Evaluate : `int_2^3 3^x dx`
Find `int (2x)/(x^2 + 1)(x^2 + 2)^2 dx`
Integration of \[\frac{1}{1 + \left( \log_e x \right)^2}\] with respect to loge x is
\[\int\frac{8x + 13}{\sqrt{4x + 7}} \text{ dx }\]
\[\int\frac{1 + x + x^2}{x^2 \left( 1 + x \right)} \text{ dx}\]
Find : \[\int\left( 2x + 5 \right)\sqrt{10 - 4x - 3 x^2}dx\] .
If θ f(x) = `int_0^x t sin t dt` then `f^1(x)` is
Find `int (dx)/sqrt(4x - x^2)`
Find: `int (dx)/(x^2 - 6x + 13)`
`int (a^x - b^x)^2/(a^xb^x)dx` equals ______.
