to evaluate the integral $int\frac{2x^{3}dx}{sqrt{2 + 4x^{2}}}$, which of the following formulas can be…

to evaluate the integral $int\frac{2x^{3}dx}{sqrt{2 + 4x^{2}}}$, which of the following formulas can be used?\n$int\frac{udu}{sqrt{a + bu}}=\frac{2}{3b^{2}}(bu - 2a)sqrt{a + bu}+c$\n$int\frac{du}{sqrt{a^{2}+u^{2}}}=ln(u+sqrt{u^{2}+a^{2}})+c$\n$int\frac{u^{2}du}{sqrt{a + bu}}=\frac{2}{15b^{3}}(3b^{2}u^{2}-4abu + 8a^{2})sqrt{a + bu}$\nwhat is the function $u$?\nanswer: $u =$\nwhat is $du$?\nanswer: $du =$ $dx$\nfind the value(s) of $a$ and $b$ (if appropriate), and use the formula to evaluate the integral. use the letter c for an arbitrary constant.\nanswer: $int\frac{2x^{3}dx}{sqrt{2 + 4x^{2}}}=$

to evaluate the integral $int\frac{2x^{3}dx}{sqrt{2 + 4x^{2}}}$, which of the following formulas can be used?\n$int\frac{udu}{sqrt{a + bu}}=\frac{2}{3b^{2}}(bu - 2a)sqrt{a + bu}+c$\n$int\frac{du}{sqrt{a^{2}+u^{2}}}=ln(u+sqrt{u^{2}+a^{2}})+c$\n$int\frac{u^{2}du}{sqrt{a + bu}}=\frac{2}{15b^{3}}(3b^{2}u^{2}-4abu + 8a^{2})sqrt{a + bu}$\nwhat is the function $u$?\nanswer: $u =$\nwhat is $du$?\nanswer: $du =$ $dx$\nfind the value(s) of $a$ and $b$ (if appropriate), and use the formula to evaluate the integral. use the letter c for an arbitrary constant.\nanswer: $int\frac{2x^{3}dx}{sqrt{2 + 4x^{2}}}=$

Answer

Explanation:

Step1: Identify the substitution

Let $u = 2x^{2}$, then $du=4xdx$.

Step2: Rewrite the integral

We have $\int\frac{2x^{3}dx}{\sqrt{2 + 4x^{2}}}=\frac{1}{2}\int\frac{u du}{\sqrt{2 + 2u}}$. Here $a = 2$ and $b = 2$. The relevant formula is $\int\frac{u du}{\sqrt{a+bu}}=\frac{2}{3b^{2}}(bu - 2a)\sqrt{a + bu}+C$.

Step3: Apply the formula

Substitute $a = 2$ and $b = 2$ into $\frac{2}{3b^{2}}(bu - 2a)\sqrt{a + bu}+C$. We get $\frac{2}{3\times2^{2}}(2u-4)\sqrt{2 + 2u}+C=\frac{1}{6}(2u - 4)\sqrt{2(1 + u)}+C$. Substitute back $u = 2x^{2}$: $\frac{1}{6}(4x^{2}-4)\sqrt{2(1 + 2x^{2})}+C=\frac{2}{3}(x^{2}-1)\sqrt{2 + 4x^{2}}+C$.

Answer:

  • The formula $\int\frac{u du}{\sqrt{a+bu}}=\frac{2}{3b^{2}}(bu - 2a)\sqrt{a + bu}+C$ can be used.
  • $u = 2x^{2}$
  • $du=4xdx$
  • $\int\frac{2x^{3}dx}{\sqrt{2 + 4x^{2}}}=\frac{2}{3}(x^{2}-1)\sqrt{2 + 4x^{2}}+C$