d) $$ \int_{0}^{\pi / 2} \int_{0}^{x^{2}} \operatorname{sen}\left(\frac{y}{x}\right) d y d x=-\frac{\pi}{2}+\frac{\pi^{2}}{8}+1 $$

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Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int_{0}^{\frac{\pi }{2}} \int_{0}^{x^{2}} \sin\left(\frac{y}{x}\right) dy dx$$
- step1: Evaluate the inner integral:
  $$\int_{0}^{\frac{\pi }{2}} \left(-x\cos\left(x\right)+x\right) dx$$
- step2: Rewrite the improper integral:
  $$\lim _{a\rightarrow 0^{+}}\left(\int_{a}^{\frac{\pi }{2}} \left(-x\cos\left(x\right)+x\right) dx\right)$$
- step3: Evaluate the integral:
  $$\lim _{a\rightarrow 0^{+}}\left(a\sin\left(a\right)+\frac{\pi ^{2}-4\pi }{8}-\frac{1}{2}a^{2}+\cos\left(a\right)\right)$$
- step4: Rewrite the expression:
  $$\lim _{a\rightarrow 0^{+}}\left(a\sin\left(a\right)\right)+\lim _{a\rightarrow 0^{+}}\left(\frac{\pi ^{2}-4\pi }{8}\right)+\lim _{a\rightarrow 0^{+}}\left(-\frac{1}{2}a^{2}\right)+\lim _{a\rightarrow 0^{+}}\left(\cos\left(a\right)\right)$$
- step5: Calculate:
  $$0+\frac{\pi ^{2}-4\pi }{8}+0+1$$
- step6: Calculate:
  $$\frac{\pi ^{2}-4\pi }{8}+0+1$$
- step7: Calculate:
  $$\frac{\pi ^{2}-4\pi }{8}+1$$
- step8: Calculate:
  $$\frac{\pi ^{2}-4\pi +8}{8}$$
Para resolver la integral doble $$ \int_{0}^{\frac{\pi}{2}} \int_{0}^{x^{2}} \sin\left(\frac{y}{x}\right) dy dx $$, hemos encontrado que el resultado es:

$$
\frac{\pi^{2} - 4\pi + 8}{8}
$$

Ahora, comparando este resultado con la expresión dada en el problema:

$$
-\frac{\pi}{2} + \frac{\pi^{2}}{8} + 1
$$

Vamos a simplificar la expresión dada para ver si son equivalentes.

Primero, reescribimos la expresión dada:

$$
-\frac{\pi}{2} + \frac{\pi^{2}}{8} + 1 = \frac{-4\pi + \pi^{2} + 8}{8}
$$

Ahora, igualamos ambas expresiones:

1. $$ \frac{\pi^{2} - 4\pi + 8}{8} $$
2. $$ \frac{-4\pi + \pi^{2} + 8}{8} $$

Ambas expresiones son iguales, lo que confirma que:

$$
\int_{0}^{\frac{\pi}{2}} \int_{0}^{x^{2}} \sin\left(\frac{y}{x}\right) dy dx = -\frac{\pi}{2} + \frac{\pi^{2}}{8} + 1
$$

Por lo tanto, la afirmación es correcta.
La integral doble $$ \int_{0}^{\frac{\pi}{2}} \int_{0}^{x^{2}} \sin\left(\frac{y}{x}\right) dy dx $$ es igual a $$ -\frac{\pi}{2} + \frac{\pi^{2}}{8} + 1 $$.

d) \( \int_{0}^{\pi / 2} \int_{0}^{x^{2}} \operatorname{sen}\left(\frac{y}{x}\right) d y d x=-\frac{\pi}{2}+\frac{\pi^{2}}{8}+1 \)

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