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Question
Find `intsqrtx/sqrt(a^3-x^3)dx`
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.
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