find the integral.\
$ \int \ 12\sqrt{x} \ dx $\
$ \int \ 12\sqrt{x} \ dx = \square $

Question

Accepted Answer

# Explanation:
## Step1: Rewrite the integrand
Rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$, so the integral becomes $\int 12x^{\frac{1}{2}}dx$.
## Step2: Use the constant multiple rule
The constant multiple rule states that $\int kf(x)dx = k\int f(x)dx$ (where $k$ is a constant). So we can factor out the 12: $12\int x^{\frac{1}{2}}dx$.
## Step3: Apply the power rule for integration
The power rule for integration is $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ (for $n
eq - 1$). Here $n=\frac{1}{2}$, so:
$$
\begin{align*}
12\int x^{\frac{1}{2}}dx&=12\times\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C\
&=12\times\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+C\
&=12\times\frac{2}{3}x^{\frac{3}{2}}+C\
& = 8x^{\frac{3}{2}}+C
\end{align*}
$$

# Answer:
$8x^{\frac{3}{2}}+C$ (or $8\sqrt{x^{3}}+C$)

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QUESTION IMAGE

Question

Explanation:

Step1: Rewrite the integrand

Step2: Use the constant multiple rule

Step3: Apply the power rule for integration

Answer: