Question

evaluate the following integral:
$\int 2x^2 - 4 \\, dx$
select one:
○ a. $4x$
○ b. $\frac{2}{3}x^3 - 4x + c$
○ c. $2x^3 - 4x + c$
○ d. $x^3 - 4x + c$
given matrices, a and b, what would the dimensions of the product matrix, ab, be?
type your answer in the form: # x #
$a = \

$$\begin{bmatrix} 2 & 1 & 7 \\\\ -8 & 0 & 2 \\end{bmatrix}$$

$ $b = \

$$\begin{bmatrix} 2 \\\\ 2 \\\\ 1 \\end{bmatrix}$$

$
answer: blank

Explanation:

First Sub - Question (Integral Evaluation)

Step 1: Recall the power rule for integration

The power rule for integration states that $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ (where $n
eq - 1$) and $\int kdx=kx + C$ (where $k$ is a constant). We will integrate each term of the function $2x^{2}-4$ separately.

Step 2: Integrate the first term $2x^{2}$

Using the power rule, for the term $2x^{2}$, we have $\int2x^{2}dx=2\times\frac{x^{2 + 1}}{2+1}=\frac{2x^{3}}{3}=\frac{2}{3}x^{3}$.

Step 3: Integrate the second term $- 4$

For the term $-4$, using the rule for integrating a constant, we have $\int(-4)dx=-4x$.

Step 4: Combine the results and add the constant of integration

When we integrate a function, we add a constant of integration $C$ (or $c$ in the options). So, $\int(2x^{2}-4)dx=\frac{2}{3}x^{3}-4x + c$.

Step 1: Recall the rule for matrix multiplication dimensions

If we have a matrix $A$ with dimensions $m\times n$ (where $m$ is the number of rows and $n$ is the number of columns) and a matrix $B$ with dimensions $n\times p$, then the product matrix $AB$ will have dimensions $m\times p$.

Step 2: Determine the dimensions of matrix $A$ and matrix $B$

Matrix $A=

$$\begin{bmatrix}2&1&7\\- 8&0&2\end{bmatrix}$$

$ has 2 rows and 3 columns, so the dimensions of $A$ are $2\times3$. Matrix $B=

$$\begin{bmatrix}2\\2\\1\end{bmatrix}$$

$ has 3 rows and 1 column, so the dimensions of $B$ are $3\times1$.

Step 3: Calculate the dimensions of $AB$

Using the rule from step 1, since $m = 2$, $n=3$ and $p = 1$, the dimensions of $AB$ will be $2\times1$.

Answer:

b. $\frac{2}{3}x^{3}-4x + c$

Second Sub - Question (Matrix Multiplication Dimensions)