Advertisements
Advertisements
Question
Integrate the function:
`1/(sqrt(x+a) + sqrt(x+b))`
Advertisements
Solution
Let I = `int 1/(sqrt(x + a) + sqrt(x + b))`
On multiplying the numerator and denominator by `sqrt(x + a) + sqrt(x + b)`
`= int 1/(sqrt(x + a) + sqrt(x + b)) xx (sqrt(x + a) - sqrt(x + b))/(sqrt(x + a) - sqrt(x + b))`
`= int (sqrt(x + a) - sqrt(x + b))/(sqrt(x + a) - sqrt(x + b))`
`= int (sqrt(x + a) - sqrt(x + b))/(a - b)` dx
`= 1/(a - b) int (x + a)^(1/2) dx - 1/(a - b) int (x + b)^(1/2)` dx
`= 1/(a - b) [2/3 (x + a)^(3/2) - 2/3 (x + b)^(3/2)] + C`
`= 2/(3(a - b))[(x + a)^(3/2) - (x + b)^(3/2)] + C`
APPEARS IN
RELATED QUESTIONS
Write the antiderivative of `(3sqrtx+1/sqrtx).`
If `f(x) =∫_0^xt sin t dt` , then write the value of f ' (x).
Find an anti derivative (or integral) of the following function by the method of inspection.
sin 2x
Find an antiderivative (or integral) of the following function by the method of inspection.
sin 2x – 4 e3x
Find the following integrals:
`int(2x^2 + e^x)dx`
Find the following integrals:
`int (x^3 + 5x^2 -4)/x^2 dx`
Find the following integrals:
`int (x^3 + 3x + 4)/sqrtx dx`
Find the following integrals:
`int (x^3 - x^2 + x - 1)/(x - 1) dx`
Find the following integrals:
`int(1 - x) sqrtx dx`
Find the following integrals:
`int(2x - 3cos x + e^x) dx`
Find the following integrals:
`int(2x^2 - 3sinx + 5sqrtx) dx`
Find the following integrals:
`int(sec^2x)/(cosec^2x) dx`
Find the following integrals:
`int (2 - 3 sinx)/(cos^2 x) dx.`
If `d/dx f(x) = 4x^3 - 3/x^4` such that f(2) = 0, then f(x) is ______.
Integrate the function:
`1/(x^2(x^4 + 1)^(3/4))`
Integrate the function:
`(5x)/((x+1)(x^2 +9))`
Integrate the function:
`(e^(5log x) - e^(4log x))/(e^(3log x) - e^(2log x))`
Integrate the function:
`cos x/sqrt(4 - sin^2 x)`
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:
`e^(3log x) (x^4 + 1)^(-1)`
Integrate the function:
f' (ax + b) [f (ax + b)]n
Integrate the function:
`sqrt((1-sqrtx)/(1+sqrtx))`
Integrate the function:
`(x^2 + x + 1)/((x + 1)^2 (x + 2))`
Evaluate `int(x^3+5x^2 + 4x + 1)/x^2 dx`
Find : \[\int\frac{\left( x^2 + 1 \right)\left( x^2 + 4 \right)}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)}dx\] .
Evaluate: `int (1 - cos x)/(cos x(1 + cos x)) dx`
`int (sin^2x - cos^2x)/(sin^2x cos^2x) dx` is equal to
What is anti derivative of `e^(2x)`
`d/(dx)x^(logx)` = ______.
`int (dx)/sqrt(5x - 6 - x^2)` equals ______.
Anti-derivative of `(tanx - 1)/(tanx + 1)` with respect to x is ______.
