What are the values of p which satisfy the quantities `int_0^p (3x^2+4x-5)dx=p^3-2`

Options

A. `1//2 and 2`
B. `-1//2 and 2`
C. `1//2 and -2`
D. `-1//2 and -2`

Answer

To solve the equation

\[ \int_0^p (3x^2 + 4x - 5) \, dx = p^3 - 2, \]

we will first compute the left-hand side integral.

### Step 1: Compute the integral

We can separate the integral into three parts:

\[ \int_0^p (3x^2 + 4x - 5) \, dx = \int_0^p 3x^2 \, dx + \int_0^p 4x \, dx - \int_0^p 5 \, dx. \]

### Step 2: Evaluate each integral

1. **Evaluate \(\int_0^p 3x^2 \, dx\)**:
\[ \int 3x^2 \, dx = x^3 \quad \text{(evaluated from 0 to p)} \]
\[ = p^3 - 0^3 = p^3. \]

2. **Evaluate \(\int_0^p 4x \, dx\)**:
\[ \int 4x \, dx = 2x^2 \quad \text{(evaluated from 0 to p)} \]
\[ = 2p^2 - 0^2 = 2p^2. \]

3. **Evaluate \(\int_0^p 5 \, dx\)**:
\[ \int 5 \, dx = 5x \quad \text{(evaluated from 0 to p)} \]
\[ = 5p - 0 = 5p. \]

### Step 3: Combine the results

Now, substituting back into the integral:

\[ \int_0^p (3x^2 + 4x - 5) \, dx = p^3 + 2p^2 - 5p. \]

### Step 4: Set the equation

Now we set this equal to the right-hand side of the original equation:

\[ p^3 + 2p^2 - 5p = p^3 - 2. \]

### Step 5: Simplify the equation

Subtract \(p^3\) from both sides:

\[ 2p^2 - 5p = -2. \]

Rearranging gives:

\[ 2p^2 - 5p + 2 = 0. \]

### Step 6: Factor the quadratic equation

To factor \(2p^2 - 5p + 2\), we look for two numbers that multiply to \(2 \times 2 = 4\) and add to \(-5\). These numbers are \(-4\) and \(-1\).

Thus, we can rewrite the equation as:

\[ 2p^2 - 4p - p + 2 = 0. \]

Grouping gives:

\[ 2p(p - 2) - 1(p - 2) = 0. \]

Factoring out \((p - 2)\):

\[ (2p - 1)(p - 2) = 0. \]

### Step 7: Solve for \(p\)

Setting each factor to zero gives:

1. \(2p - 1 = 0 \Rightarrow p = \frac{1}{2}\)
2. \(p - 2 = 0 \Rightarrow p = 2\)

### Final Answer

The values of \(p\) that satisfy the equation are:

\[ p = \frac{1}{2} \quad \text{and} \quad p = 2. \]

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