What is the result of the integral ∫ (3 + 4x) dx?

What is the result of the integral ∫ (3 + 4x) dx?

• A. 3x + 2x^2 + C
• B. 4x + 2x^2 + C
• C. 3x + 4x^2 + C
• D. 4x + 3x^2 + C

Answer: (A)

To solve the integral ∫ (3 + 4x) dx, we apply the basic rules of integration. First, notice that the integral is composed of two separate terms: a constant (3) and a linear function (4x). We can integrate each term individually. The integral of a constant 'a' is given by \( ax \), so the integral of the constant term 3 will be: \[ \int 3 \, dx = 3x \] Next, we move on to the term involving 'x'. The integral of a linear function \( ax \) is given by \( \frac{a}{2}x^2 \), thus for the term 4x, we have: \[ \int 4x \, dx = 4 \cdot \frac{x^2}{2} = 2x^2 \] Now, we combine the results from both parts of the integral: \[ \int (3 + 4x) \, dx = 3x + 2x^2 + C \] where C represents the constant of integration, which is included because this is an indefinite integral. The correct result of the integral is

To solve the integral ∫ (3 + 4x) dx, we apply the basic rules of integration.

First, notice that the integral is composed of two separate terms: a constant (3) and a linear function (4x). We can integrate each term individually.

The integral of a constant 'a' is given by ( ax ), so the integral of the constant term 3 will be:

[

\int 3 , dx = 3x

]

Next, we move on to the term involving 'x'. The integral of a linear function ( ax ) is given by ( \frac{a}{2}x^2 ), thus for the term 4x, we have:

[

\int 4x , dx = 4 \cdot \frac{x^2}{2} = 2x^2

]

Now, we combine the results from both parts of the integral:

[

\int (3 + 4x) , dx = 3x + 2x^2 + C

]

where C represents the constant of integration, which is included because this is an indefinite integral.

The correct result of the integral is