Mathematics, 25.12.2021 22:30 leannamat2106

Find g(x) if the indefinite integral of f(x) need little help

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Answer from: makaylahamrick

g(x) = 4x² - x

General Formulas and Concepts:

Algebra II

Functions

Function NotationPiecewise Functions

Calculus

Integration

IntegralsIntegral NotationIntegration Constant C

Integration Rule [Reverse Power Rule]: $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Property [Multiplied Constant]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

Integration Property [Addition/Subtraction]: $\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx$

Step-by-step explanation:

*Note:

Integrating a piecewise function requires you to integrate both parts.

Step 1: Define

Identify.

$\displaystyle f(x) = \left \{ {{8x - 1 ,\ x \leq 4} \atop {31 ,\ x \geq 4}} \right.$

$\displaystyle \int {f(x)} \, dx = \left \{ {{g(x) + C ,\ x \leq 4} \atop {31x + C ,\ x \geq 4}} \right.$

Step 2: Find function g(x)

We can see that the 2nd part of the piecewise function already has been integrated:

[Integral] Set up: $\displaystyle \int {f(x)} \, dx ,\ x \geq 4 = \int {31} \, dx ,\ x \geq 4$ [Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {f(x)} \, dx ,\ x \geq 4 = 31 \int {} \, dx ,\ x \geq 4$ [Integral] Integrate [Integration Rule - Reverse Power Rule]: $\displaystyle \int {f(x)} \, dx ,\ x \geq 4 = 31x + C ,\ x \geq 4$

To find function g(x), we simply have the same setup:

[Integral] Set up: $\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = \int {8x - 1} \, dx ,\ x \leq 4$ [Integral] Rewrite [Integration Rule - Addition/Subtraction]: $\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = \int {8x} \, dx - \int {1} \, dx ,\ x \leq 4$ [Integrals] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = 8 \int {x} \, dx - \int {} \, dx ,\ x \leq 4$ [Integrals] Integrate [Integration Rule - Reverse Power Rule]: $\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = 8 \bigg( \frac{x^2}{2} \bigg) - x + C ,\ x \leq 4$ Simplify: $\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = 4x^2 - x + C ,\ x \leq 4$ Redefine: $\displaystyle g(x) = 4x^2 - x + C ,\ x \leq 4$

The integration constant C is already included in the answer, so our answer is g(x) = 4x² - x.

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration (Applications)

Answer from: Quest

1.4

2.3

step-by-step explanation:

the formula for the constant of variation is

y= kx

divide each side by x

y/x =kx/x

y/x =k

we know a and y

1. y/x = k

4/1 =4

check the other values to make sure they are the same

8/2 = 4 12/3 =4 and 16/4 =4

the constant of variation is 4

2. y/x = k

3/1 =3

check the other values to make sure they are the same

6/2 = 3 9/3 =3 and 12/4 =3

the constant of variation is 3

Answer from: Quest

yess

step-by-step explanation:

idk whyy this is happening, i cant even read my notification bc i keep getting errors

~coco

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