Given answer for Exercise 6 (1)a: F(x) = 4x³ + 3x² − 5x + c.

Question

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I will assume you want me to demonstrate the integration process for (1)a. This means I will find the function whose indefinite integral is given by the expression in (1)a, and then integrate that function to verify the result.

Given answer for (1)a: $F(x) = 4x^3 + 3x^2 - 5x + c$ .

Determine the function $f(x)$ that was integrated to obtain $F(x)$ .
To do this, we find the derivative of $F(x)$ .
$f(x) = \frac{d}{dx}(4x^3 + 3x^2 - 5x + c)$ $f(x) = 4 \cdot 3x^{3-1} + 3 \cdot 2x^{2-1} - 5 \cdot 1x^{1-1} + 0$ $f(x) = 12x^2 + 6x - 5$
Integrate $f(x)$ to verify the given answer.
We need to find $\int (12x^2 + 6x - 5)\,dx$ .
Using the power rule for integration, $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ :
$\int (12x^2 + 6x - 5)\,dx = \int 12x^2\,dx + \int 6x\,dx - \int 5\,dx$ $= 12 \left(\frac{x^{2+1}}{2+1}\right) + 6 \left(\frac{x^{1+1}}{1+1}\right) - 5x + c$ $= 12 \left(\frac{x^3}{3}\right) + 6 \left(\frac{x^2}{2}\right) - 5x + c$ $= 4x^3 + 3x^2 - 5x + c$
The result matches the given expression.

The verified indefinite integral is:
$\mathbf{4x^3 + 3x^2 - 5x + c}$

Given answer for Exercise 6 (1)a: F(x) = 4x³ + 3x² − 5x + c.

Answer

Step-by-step solution

Determine the function $f(x)$ that was integrated to obtain $F(x)$ .

Integrate $f(x)$ to verify the given answer.

Try one yourself

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Given answer for Exercise 6 (1)a: F(x) = 4x³ + 3x² − 5x + c.

Answer

Step-by-step solution

Determine the function f(x)f(x)f(x) that was integrated to obtain F(x)F(x)F(x).

Integrate f(x)f(x)f(x) to verify the given answer.

Try one yourself

Still stuck on this one?

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Determine the function $f(x)$ that was integrated to obtain $F(x)$ .

Integrate $f(x)$ to verify the given answer.