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
Write the antiderivative of `(3sqrtx+1/sqrtx).`
Find :`int(x^2+x+1)/((x^2+1)(x+2))dx`
If `f(x) =∫_0^xt sin t dt` , then write the value of f ' (x).
Find an antiderivative (or integral) of the following function by the method of inspection.
sin 2x – 4 e3x
Find the following integrals:
`int (4e^(3x) + 1)`
Find the following integrals:
`intx^2 (1 - 1/x^2)dx`
Find the following integrals:
`int(2x^2 + e^x)dx`
Find the following integrals:
`intsec x (sec x + tan x) dx`
Find the following integrals:
`int(sec^2x)/(cosec^2x) dx`
Find the following integrals:
`int (2 - 3 sinx)/(cos^2 x) dx.`
The anti derivative of `(sqrtx + 1/ sqrtx)` equals:
Integrate the function:
`1/(x - x^3)`
Integrate the function:
`1/(sqrt(x+a) + sqrt(x+b))`
Integrate the function:
`1/(x^2(x^4 + 1)^(3/4))`
Integrate the function:
`(sin^8 x - cos^8 x)/(1-2sin^2 x cos^2 x)`
Integrate the function:
`e^x/((1+e^x)(2+e^x))`
Integrate the function:
`1/((x^2 + 1)(x^2 + 4))`
Integrate the function:
`cos^3 xe^(log sinx)`
Integrate the function:
`(x^2 + x + 1)/((x + 1)^2 (x + 2))`
Integrate the function:
`tan^(-1) sqrt((1-x)/(1+x))`
Integrate the function:
`(sqrt(x^2 +1) [log(x^2 + 1) - 2log x])/x^4`
Evaluate: `int (1 - cos x)/(cos x(1 + cos x)) dx`
If `d/(dx) f(x) = 4x^3 - 3/x^4`, such that `f(2) = 0`, then `f(x)` is
`int (dx)/(sin^2x cos^2x) dx` equals
`int (dx)/sqrt(9x - 4x^2)` equal
`int (dx)/sqrt(9x - 4x^2)` equals
`int (xdx)/((x - 1)(x - 2))` equals
`int sqrt(1 + x^2) dx` is equal to
`int (dx)/sqrt(5x - 6 - x^2)` equals ______.
What is integration called because it reverses the operation of finding derivatives?
What is \(C\) called in \[\int f(x)\,dx=F(x)+C\]?
Which antiderivative has derivative \(1\)?
Evaluate \[\int \sec x\tan x\,dx\]
Evaluate \[\int \frac{1}{x}\,dx\]
Evaluate \[\int a^x\,dx\]
