# Find ∫√x/√(a^3−x^3)dx - Mathematics and Statistics

Find intsqrtx/sqrt(a^3-x^3)dx

#### Solution

I=intsqrtx/sqrt(a^3-x^3)dx

Letx^(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.

Concept: Methods of Integration: Integration by Substitution
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