Question

integrtion from 0 to a x2(a-x)/(a+x)dx​

Answer 1

Answer:

The best substitution for this integral is

x=acos2θ+bsin2θ

Then we have the following.

x−a=acos2θ+bsin2θ−a=bsin2θ−a(1−cos2θ)

=bsin2θ−asin2θ=(b−a)sin2θ

Similarly,

b−x=(b−a)cos2θ

And,

dx=−2asinθcosθ+2bcosθsinθdθ

=2(b−a)sinθcosθdθ

We now have

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Step-by-step explanation:

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Answer 2

First of all you transform [math]ax-x^2[/math] into [math]c^2-(x-b)^2.[/math] (Express [math]b, c[/math] in [math]a[/math]). Then you remove [math]c^2[/math] from the sqrt sign and let

[math]\displaystyle t=\frac{x-b}c[/math]

This should result in a familiar expression to integrate. This is standard practice in integrals with squares of the variable under a sqrt sign in the denominator: by shifting and scaling you bring them back to the familiar form.