3. Resolver la siguiente integral. (Valor 1.) $$ \int x^{3} \sqrt{2 x^{4}-3} d x $$

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Calculate the integral $$ \int x^{3} \sqrt{2x^{4}-3} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int x^{3}\sqrt{2x^{4}-3} dx$$
- step1: Rewrite the expression:
  $$\int x^{3}\left(2x^{4}-3ight)^{\frac{1}{2}} dx$$
- step2: Use the substitution $$dx=\frac{1}{8x^{3}} dt$$ to transform the integral$$:$$
  $$\int x^{3}\left(2x^{4}-3ight)^{\frac{1}{2}}	imes \frac{1}{8x^{3}} dt$$
- step3: Simplify:
  $$\int \frac{\left(2x^{4}-3ight)^{\frac{1}{2}}}{8} dt$$
- step4: Use the substitution $$t=2x^{4}$$ to transform the integral$$:$$
  $$\int \frac{\left(t-3ight)^{\frac{1}{2}}}{8} dt$$
- step5: Rewrite the expression:
  $$\int \frac{1}{8}\left(t-3ight)^{\frac{1}{2}} dt$$
- step6: Use properties of integrals:
  $$\frac{1}{8}	imes \int \left(t-3ight)^{\frac{1}{2}} dt$$
- step7: Use the substitution $$dt=1 dv$$ to transform the integral$$:$$
  $$\frac{1}{8}	imes \int \left(t-3ight)^{\frac{1}{2}}	imes 1 dv$$
- step8: Simplify:
  $$\frac{1}{8}	imes \int \left(t-3ight)^{\frac{1}{2}} dv$$
- step9: Use the substitution $$v=t-3$$ to transform the integral$$:$$
  $$\frac{1}{8}	imes \int v^{\frac{1}{2}} dv$$
- step10: Evaluate the integral:
  $$\frac{1}{8}	imes \frac{v^{\frac{1}{2}+1}}{\frac{1}{2}+1}$$
- step11: Simplify:
  $$\frac{1}{8}	imes \frac{v^{\frac{3}{2}}}{\frac{3}{2}}$$
- step12: Multiply the terms:
  $$\frac{v^{\frac{3}{2}}}{8	imes \frac{3}{2}}$$
- step13: Multiply the terms:
  $$\frac{v^{\frac{3}{2}}}{12}$$
- step14: Substitute back:
  $$\frac{\left(t-3ight)^{\frac{3}{2}}}{12}$$
- step15: Substitute back:
  $$\frac{\left(2x^{4}-3ight)^{\frac{3}{2}}}{12}$$
- step16: Simplify:
  $$\frac{\sqrt{\left(2x^{4}-3ight)^{3}}}{12}$$
- step17: Add the constant of integral C:
  $$\frac{\sqrt{\left(2x^{4}-3ight)^{3}}}{12} + C, C \in \mathbb{R}$$
La integral de $$ \int x^{3} \sqrt{2x^{4}-3} dx $$ es $$ \frac{\sqrt{(2x^{4}-3)^{3}}}{12} + C $$, donde $$ C $$ es una constante real.
La integral es $$ \frac{\sqrt{(2x^{4}-3)^{3}}}{12} + C $$.

Resolver la siguiente integral. (Valor 1.)

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