Advertisements
Advertisements
प्रश्न
Integrate the function:
`1/(xsqrt(ax - x^2)) ["Hint : Put x" = a/t]`
Advertisements
उत्तर
Let `1/(xsqrt(ax - x^2))`
Put `x = a/t`
dx = `- a/t^2 dt`
Now, `xsqrt(ax - x^2) = a/tsqrt(a xx a/t - a^2/t^2)`
`= a^2/t sqrt(1/t - 1/t^2) = a^2/t^2 sqrt(t - 1)`
`therefore I = 1/(a^2/t^2 sqrt(t - 1)) xx (- a)/t^2 dt`
`= - 1/a int 1/sqrt(t - 1) dt`
`= - 1/a ((t - 1)^(- 1/2 + 1))/(- 1/2 + 1) + C`
`= - 1/a (t - 1)^(1/2)/(1/2) + C`
`= - 2/a sqrt(t - 1) + C`
`= - 2/a sqrt(a/x - 1) + C`
`= - 2/a sqrt((a - x)/x) + C`
APPEARS IN
संबंधित प्रश्न
Find :`int(x^2+x+1)/((x^2+1)(x+2))dx`
Find an anti derivative (or integral) of the following function by the method of inspection.
sin 2x
Find an anti derivative (or integral) of the following function by the method of inspection.
(axe + b)2
Find the following integrals:
`int (ax^2 + bx + c) dx`
Find the following integrals:
`int(2x^2 + e^x)dx`
Find the following integrals:
`int(1 - x) sqrtx dx`
Find the following integrals:
`int (2 - 3 sinx)/(cos^2 x) dx.`
Integrate the function:
`1/(sqrt(x+a) + sqrt(x+b))`
Integrate the function:
`1/(x^2(x^4 + 1)^(3/4))`
Integrate the function:
`1/(x^(1/2) + x^(1/3)) ["Hint:" 1/(x^(1/2) + x^(1/3)) = 1/(x^(1/3)(1+x^(1/6))), "put x" = t^6]`
Integrate the function:
`(5x)/((x+1)(x^2 +9))`
Integrate the function:
`sinx/(sin (x - a))`
Integrate the function:
`1/((x^2 + 1)(x^2 + 4))`
Integrate the function:
`e^(3log x) (x^4 + 1)^(-1)`
Integrate the functions `(sin^(-1) sqrtx - cos^(-1) sqrtx)/ (sin^(-1) sqrtx + cos^(-1) sqrtx) , x in [0,1]`
Integrate the function:
`sqrt((1-sqrtx)/(1+sqrtx))`
Integrate the function:
`(x^2 + x + 1)/((x + 1)^2 (x + 2))`
Integrate the function:
`(sqrt(x^2 +1) [log(x^2 + 1) - 2log x])/x^4`
The anti derivative of `(sqrt(x) + 1/sqrt(x))` is equals:
`sqrt((10x^9 + 10^x log e^10)/(x^10 + 10^x)) dx` equals
What is anti derivative of `e^(2x)`
`d/(dx)x^(logx)` = ______.
In \[\int f(x)\,dx\], what is \(f(x)\) called?
What is “an integral of \(f\)” ?
What is integration?
Which statement about indefinite integrals is correct?
For \(n\ne -1\), which is the correct integral of \(x^n\)?
Which antiderivative has derivative \(1\)?
Evaluate \[\int \sin x\,dx\]
Evaluate \[\int \text{cosec}^2 x\,dx\]
Evaluate \[\int \text{cosec} x\cot x\,dx\]
Which expression is an antiderivative of \[\frac{1}{\sqrt{1-x^2}}\]?
Evaluate \[\int e^x\,dx\]
Evaluate \[\int a^x\,dx\]
