Question

whether the composition of two r-s integrable functions is r-s integrable? justify the answer

Answer 1

Let the two function be r=x and s=$x^{2}$

$\int x dx=\frac{x^2}{2}+ c,</p><p> \int x^2 dx=\frac{x^3}{3}+ c$

Here r and s both are integrable functions.

Now their composition (r-s) i.e x-$x^{2}$ is also integrable.

$\int(x-x^{2})dx=\frac{x^2}{2}-\frac{x^3}{3}+k$

You can take any two function , and this result is true i.e the composition of two r-s integrable functions is r-s integrable.

If composition is (-) function i.e r=x and s=$x^{2}$

then r-s(x)=r(x²)=x² , also s-r(x)=s(x)=x²

As you can see r and s both are integrable, so their composition r-s(x)

and s-r(x) is also integrable.

And this is true for any two different functions.