Math  /  Calculus

QuestionFind ff. (Use CC for the constant of the first antiderivative and DD for the constant of the second antiderivative.) f(x)=24x318x2+8xf(x)=\begin{array}{l} f(x)=24 x^{3}-18 x^{2}+8 x \\ f(x)=\square \end{array} Need Help? Read It Watch it

Studdy Solution

STEP 1

What is this asking? We're given the derivative of a function, f(x)f'(x), and we need to find the original function, f(x)f(x), by taking the antiderivative (also called integration). Watch out! Don't forget to add the constant of integration (CC) after the *first* antiderivative, and another constant (DD) after the *second*!

STEP 2

1. Find the first antiderivative.
2. Find the second antiderivative.

STEP 3

Alright, let's **kick things off** by finding the *first* antiderivative of f(x)=24x318x2+8xf'(x) = 24x^3 - 18x^2 + 8x.
This means we're going to integrate f(x)f'(x) with respect to xx.
Mathematically, we write this as: (24x318x2+8x)dx \int (24x^3 - 18x^2 + 8x) \, dx

STEP 4

Remember the **power rule for integration**: xndx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.
We add one to the exponent and then divide by the new exponent.
Don't forget our constant of integration, CC!

STEP 5

Let's apply this rule to each term in our integral: 24x3dx=24x3+13+1=24x44=6x4 \int 24x^3 \, dx = \frac{24x^{3+1}}{3+1} = \frac{24x^4}{4} = 6x^4 18x2dx=18x2+12+1=18x33=6x3 \int -18x^2 \, dx = \frac{-18x^{2+1}}{2+1} = \frac{-18x^3}{3} = -6x^3 8xdx=8x1+11+1=8x22=4x2 \int 8x \, dx = \frac{8x^{1+1}}{1+1} = \frac{8x^2}{2} = 4x^2

STEP 6

Putting it all together, the *first* antiderivative is: 6x46x3+4x2+C 6x^4 - 6x^3 + 4x^2 + C

STEP 7

Now, we need to find the antiderivative *again*!
We're integrating 6x46x3+4x2+C6x^4 - 6x^3 + 4x^2 + C with respect to xx: (6x46x3+4x2+C)dx \int (6x^4 - 6x^3 + 4x^2 + C) \, dx

STEP 8

Using the **power rule** on each term (just like before!): 6x4dx=6x55 \int 6x^4 \, dx = \frac{6x^5}{5} 6x3dx=6x44=3x42 \int -6x^3 \, dx = \frac{-6x^4}{4} = -\frac{3x^4}{2} 4x2dx=4x33 \int 4x^2 \, dx = \frac{4x^3}{3} Cdx=Cx \int C \, dx = Cx

STEP 9

Combining these results and adding our *second* constant of integration, DD, gives us the **final answer**: 6x553x42+4x33+Cx+D \frac{6x^5}{5} - \frac{3x^4}{2} + \frac{4x^3}{3} + Cx + D

STEP 10

f(x)=6x553x42+4x33+Cx+D f(x) = \frac{6x^5}{5} - \frac{3x^4}{2} + \frac{4x^3}{3} + Cx + D

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