Solve the problem involving the given double integral

[chwala]

Homework Statement

see attached. Interest is on ( Problem number 9) ... i thought its the most challenging one on the page...

Relevant Equations

Integration

Ok in my approach i have the lines,
starting with the inner integral,

$$\int_0^1 xy \cos (x^2y) dx$$

I let $u =x^2y , u(0)=0, u(1)=y$

...

$$\dfrac{1}{2} \int_0^y \cos u du=\left[\dfrac{1}{2} \sin u \right]_0^y= \left[\dfrac{1}{2} \sin (x^2y) \right]_0^1=\left[\dfrac{1}{2} \sin y \right]$$Now to the outer integral,
$$ \dfrac{1}{2} \int_0^{0.5π} \sin y dy= \left[-\dfrac {1}{2} \cos y \right]_0^{0.5π}=-0+\dfrac{1}{2}= \dfrac{1}{2}$$

Any input is welcome trying to refresh on this things...

 

[fresh_42]

Homework Statement: see attached. Interest is on ( Problem number 9) ... i thought its the most challenging one on the page...
Relevant Equations: Integration

View attachment 336749

Ok in my approach i have the lines,
starting with the inner integral,

$$\int_0^1 xy \cos (x^2y) dx$$

I let $u =x^2y , u(0)=0, u(1)=y$

...

$$\dfrac{1}{2} \int_0^y \cos u du=\left[\dfrac{1}{2} \sin u \right]_0^y= \left[\dfrac{1}{2} \sin (x^2y) \right]_0^1=\left[\dfrac{1}{2} \sin y \right]$$Now to the outer integral,
$$ \dfrac{1}{2} \int_0^{0.5π} \sin y dy= \left[-\dfrac {1}{2} \cos y \right]_0^{0.5π}=-0+\dfrac{1}{2}= \dfrac{1}{2}$$

Any input is welcome trying to refresh on this things...

Well, it is correct. I would prefer a single line of argumentation over those split equations you use. E.g.
\begin{align*}
\int_{0}^{\pi/2}\int_0^1 xy\,\cos(x^2y) \,dx \,dy&= \left.\int_{0}^{\pi/2}\int_{x=0}^{x=1} xy\,\cos(u)\,dx \,dy \quad\right| \;u:=x^2y\, , \,\dfrac{du}{dx}=2xy\, , \,xy\cdot dx=\dfrac{du}{2}\\
&=\dfrac{1}{2}\int_{0}^{\pi/2} \int_{u=0}^{u=y} \cos(u) \,du\,dy \\
&=\dfrac{1}{2}\int_{0}^{\pi/2}\left[\sin(u)\right]_0^y\;dy\\
&=\dfrac{1}{2}\int_{0}^{\pi/2} \sin(y)\,dy \\&=-\dfrac{1}{2} \left[\cos(y)\right]_{0}^{\pi/2}\\
&=-\dfrac{1}{2}\cdot (0-1)\\
&=\dfrac{1}{2}
\end{align*}
I think we should make a distinction between what we scribble down as a calculation and what we write down at the end. This has an additional advantage if things are more complicated than this. It forces you to reconsider the calculation step by step and discloses possible mistakes. As I said, this is very valuable in more complex situations, e.g. if your proof takes pages instead of lines. It's better to learn it with lines before it becomes pages.