Advertisements
Advertisements
Question
Find `intsqrtx/sqrt(a^3-x^3)dx`
Advertisements
Solution
`I=intsqrtx/sqrt(a^3-x^3)dx`
Let: `x^(3/2)=t`
`=>3/2x^(1/2)dx=dt`
`x^(1/2)dx=2/3dt`
Putting the values in I, we get
`I=intsqrtx/sqrt(a^3-x^3)dx`
`=2/3int1/(sqrt(a^3-t^2))dt`
Using the following formula of integration, we get
`intdx/sqrt(a^2-x^2)=sin^(-1)(x/a)`
`:.2/3int1/sqrt(a^3-t^2)dt=2/3sin^(-1)(t/(a^(3/2)))+C`
Again, putting the value of t, we get
`2/3int1/sqrt(a^3-t^2)dt=2/3sin^(-1)(t/a^(3/2))+C`
`=2/3sin^(-1)(x^(3/2)/a^(3/2))+C`
Here, C is constant of integration.
APPEARS IN
RELATED QUESTIONS
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Evaluate: `int sqrt(tanx)/(sinxcosx) dx`
Integrate the functions:
`(e^(2x) - 1)/(e^(2x) + 1)`
Integrate the functions:
cot x log sin x
Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`
Write a value of
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Evaluate the following integrals : `int cos^2x.dx`
Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
`int 1/(cos x - sin x)` dx = _______________
`int cot^2x "d"x`
State whether the following statement is True or False:
If `int x "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
`int(log(logx) + 1/(logx)^2)dx` = ______.
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Which substitution is appropriate for \[\sqrt{\frac{\mathrm{a}-x}{\mathrm{a}+x}}\] or \[\sqrt{\frac{\mathrm{a}+x}{\mathrm{a}-x}}\]?
