Evaluate ${}_{R}rsin(m)$ es$,$ where $R=[2,6][0,].$

Solution $1$

If we first integrate with respect so $x,$ we pet the following.

Solution $2$

If we reverse the order of integration, we get the following.

${}_{n}ysin(xy)dA={\displaystyle \underset{2}{\overset{6}{}}}{\displaystyle \underset{0}{\overset{x}{}}}ysin(xy)dydx$

To evaluate the inner integral, we use integration by parts with

$u=r,or=$

$$ dy$.$

$du=dy,v=-\frac{cos(xy)}{x}$

and $50$

${\displaystyle \underset{0}{\overset{x}{}}}ysin(xy)dy=-\frac{ycos(xy)}{x}{|}_r|=0^r=x\frac{1}{x}{\displaystyle \underset{0}{\overset{x}{}}}()$

If we now integrate the flrst term by parts with $w=-\frac{1}{x}$ and dr $=$ ds$,$ we get du $=$ dr$.v=sin(x),$ and

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}-\frac{cos(x)}{x}dx=-{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{sin(x)}{x^2}dx$

Therefore

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(-\frac{cos(x)}{x}\frac{sin(x)}{x^2})dx=$

$$

and $10$

${\displaystyle \underset{2}{\overset{4}{}}}{\displaystyle \underset{0}{\overset{}{}}}ysin(xy)dydx=$

$=$