Math / Calculus

Question1. [-/1 Points] DETAILS MY NOTES SCALCET9 4.9.006.
Find the most general antiderivative of the function. (Check your answer by differentiation. Use $C$ for the constant of the antiderivative. $\begin{array}{l} f(x)=x^{2}-7 x+8 \\ F(x)=\square \end{array}$ Need Help? Road It Submit Answer

Studdy Solution

STEP 1

1. We are given the function $f(x) = x^2 - 7x + 8$ .
2. We need to find the most general antiderivative $F(x)$ .
3. We will use $C$ as the constant of integration.

STEP 2

1. Identify the antiderivative of each term in the function separately.
2. Combine the antiderivatives to form the most general antiderivative.
3. Add the constant of integration $C$ .
4. Differentiate the antiderivative to check the solution.

STEP 3

Identify the antiderivative of each term:
- The antiderivative of $x^2$ is $\frac{x^3}{3}$ . - The antiderivative of $-7x$ is $-\frac{7x^2}{2}$ . - The antiderivative of $8$ is $8x$ .

STEP 4

Combine the antiderivatives:
$F(x) = \frac{x^3}{3} - \frac{7x^2}{2} + 8x + C$

STEP 5

Add the constant of integration $C$ :
The most general antiderivative is:
$F(x) = \frac{x^3}{3} - \frac{7x^2}{2} + 8x + C$

STEP 6

Differentiate $F(x)$ to check the solution:
$F'(x) = \frac{d}{dx} \left( \frac{x^3}{3} - \frac{7x^2}{2} + 8x + C \right)$
$F'(x) = x^2 - 7x + 8$
Since $F'(x) = f(x)$ , the solution is verified.
The most general antiderivative is:
$\boxed{\frac{x^3}{3} - \frac{7x^2}{2} + 8x + C}$

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Question1. [-/1 Points] DETAILS MY NOTES SCALCET9 4.9.006.
Find the most general antiderivative of the function. (Check your answer by differentiation. Use $C$ for the constant of the antiderivative. $\begin{array}{l} f(x)=x^{2}-7 x+8 \\ F(x)=\square \end{array}$ Need Help? Road It Submit Answer

Studdy Solution

STEP 1

1. We are given the function $f(x) = x^2 - 7x + 8$ .
2. We need to find the most general antiderivative $F(x)$ .
3. We will use $C$ as the constant of integration.

STEP 2

1. Identify the antiderivative of each term in the function separately.
2. Combine the antiderivatives to form the most general antiderivative.
3. Add the constant of integration $C$ .
4. Differentiate the antiderivative to check the solution.

STEP 3

Identify the antiderivative of each term:
- The antiderivative of $x^2$ is $\frac{x^3}{3}$ . - The antiderivative of $-7x$ is $-\frac{7x^2}{2}$ . - The antiderivative of $8$ is $8x$ .

STEP 4

Combine the antiderivatives:
$F(x) = \frac{x^3}{3} - \frac{7x^2}{2} + 8x + C$

STEP 5

Add the constant of integration $C$ :
The most general antiderivative is:
$F(x) = \frac{x^3}{3} - \frac{7x^2}{2} + 8x + C$

STEP 6

Differentiate $F(x)$ to check the solution:
$F'(x) = \frac{d}{dx} \left( \frac{x^3}{3} - \frac{7x^2}{2} + 8x + C \right)$
$F'(x) = x^2 - 7x + 8$
Since $F'(x) = f(x)$ , the solution is verified.
The most general antiderivative is:
$\boxed{\frac{x^3}{3} - \frac{7x^2}{2} + 8x + C}$

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Studdy Solution

STEP 1

1. We are given the function f(x)=x2−7x+8 f(x) = x^2 - 7x + 8 f(x)=x2−7x+8.2. We need to find the most general antiderivative F(x) F(x) F(x).3. We will use C C C as the constant of integration.

STEP 2

1. Identify the antiderivative of each term in the function separately.2. Combine the antiderivatives to form the most general antiderivative.3. Add the constant of integration C C C.4. Differentiate the antiderivative to check the solution.

STEP 3

Identify the antiderivative of each term:- The antiderivative of x2 x^2 x2 is x33 \frac{x^3}{3} 3x3​. - The antiderivative of −7x -7x −7x is −7x22 -\frac{7x^2}{2} −27x2​. - The antiderivative of 8 8 8 is 8x 8x 8x.

STEP 4

Combine the antiderivatives:F(x)=x33−7x22+8x+C F(x) = \frac{x^3}{3} - \frac{7x^2}{2} + 8x + C F(x)=3x3​−27x2​+8x+C

STEP 5

Add the constant of integration C C C:The most general antiderivative is:F(x)=x33−7x22+8x+C F(x) = \frac{x^3}{3} - \frac{7x^2}{2} + 8x + C F(x)=3x3​−27x2​+8x+C

STEP 6

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1. We are given the function $f(x) = x^2 - 7x + 8$ .
2. We need to find the most general antiderivative $F(x)$ .
3. We will use $C$ as the constant of integration.

1. Identify the antiderivative of each term in the function separately.
2. Combine the antiderivatives to form the most general antiderivative.
3. Add the constant of integration $C$ .
4. Differentiate the antiderivative to check the solution.

Identify the antiderivative of each term:
- The antiderivative of $x^2$ is $\frac{x^3}{3}$ . - The antiderivative of $-7x$ is $-\frac{7x^2}{2}$ . - The antiderivative of $8$ is $8x$ .

Combine the antiderivatives:
$F(x) = \frac{x^3}{3} - \frac{7x^2}{2} + 8x + C$

Add the constant of integration $C$ :
The most general antiderivative is:
$F(x) = \frac{x^3}{3} - \frac{7x^2}{2} + 8x + C$