Question

Find the particular antiderivative of the following derivative that satisfies the given condition

C'(x)=4X^2-3X C(0)=1,000

Answer 1

@jim_thompson5910 you know you want to help me! :)

Answer 2

First integrate both sides with respect to x


\[\large C^{\prime}(x) = 4x^2 - 3x\]


\[\large \int C^{\prime}(x)dx = \int (4x^2 - 3x)dx\]


\[\large C(x) = \frac{4}{3}x^3-\frac{3}{2}x^2+D\]


Note: I used D as the constant since C is used in the function name

Answer 3

now we use the fact that C(0) = 1000 to find D

\[\large C(x) = \frac{4}{3}x^3-\frac{3}{2}x^2+D\]

\[\large C(0) = \frac{4}{3}(0)^3-\frac{3}{2}(0)^2+D\]

\[\large 1000 = \frac{4}{3}(0)^3-\frac{3}{2}(0)^2+D\]

\[\large 1000 = \frac{4}{3}(0)-\frac{3}{2}(0)+D\]

\[\large 1000 = 0-0+D\]

\[\large 1000 = D\]

\[\large D = 1000\]

So the function C(x) is

\[\large C(x) = \frac{4}{3}x^3-\frac{3}{2}x^2+1000\]

Answer 4

How do you know how to integrate it? Is it different for each indefinite integral problem

Answer 5

when it says "find the antiderivative", that's the essentially the same as saying "integrate"

Answer 6

the antiderivative is more descriptive because it literally means "opposite of derivative", so you're undoing the derivative....which is exactly what an indefinite integral is

Answer 7

Can I pick your brain in a different example? And see how that differs?

Answer 8

So for example

say we have

f(x) = x^2

we derive it to get

f ' (x) = 2x

-------------

now if we integrate (aka take the antiderivative), we would get

integral of ( f ' (x) ) = x^2 + C

so we kinda undid what the derivative did to x^2 to make it 2x...but there's that constant added to it (since the derivative of x^2 and x^2 + 1 is the same)

Answer 9

sure

Answer 10

Find the indefinite integral 1/2+5x^6 all times (30x^5) dx

Answer 11

so it's

\[\large \left(\frac{1}{2}+5x^6\right)(30x^5)\]

right?

Answer 12

or is it

\[\large \left(\frac{1}{2+5x^6}\right)(30x^5)\]