Question

How do I find an indefinite integral

Answer 1

That depends on the integral. In general, we consider integration as taking an infinite sum of very small elements.

Answer 2

In some cases, you can take the definite integral, and let the limit of integration approach infinity (or negative infinity) as required.

Answer 3

well this is the problem ∫[-a,9] [x^2+x]

Answer 4

im not even sure where to begin

Answer 5

You mean, like this: \[ \int_{-a}^{9} (x^2 + x) dx \]?

Answer 6

yes

Answer 7

In this case, just take the integral and evaluate at nine and -a.

Answer 8

Well, let us focus on the antiderivative first. Break it into two parts: \[\int x^2dx + \int xdx \]Can you work this out?

Answer 9

my teacher did not explain it well i'm not sure how to do it

Answer 10

Ok then. There are a couple of rules you have to bear in mind. The simplest one is this: \[\int x^ndx = \frac{x^{n+1}}{n+1}\]This is called the power rule for integration.

Answer 11

so the first is x^3/3?

Answer 12

That comes from the fact that, when we take the antiderivative, we are looking for another function F(x) whose derivative will argument that we integrated. That is, what is \[\LARGE\frac{d(\frac{x^{n+1}}{n+1})}{dx} =?\]

Answer 13

Yeah, it's :-)

Answer 14

Typo: whose derivative will be the argument that we previously integrated.

Answer 15

and the other one is x^2/2

Answer 16

Yup. So we get: \[\int (x^2 + x)dx = \frac{x^3}{3} + \frac{x^2}{2} \]

Answer 17

But now we have to work out the limits.

Answer 18

\[\int\limits_{a}^{b}f(x)dx = F(b) - F(a)\]That is, after we computed F (from the antiderivative) we need to plug in the values for b and a and simplify,

Answer 19

The end result of a definite integral should be a number, or something that recalls a number. Not a function, as oppose to the indefinite integral.

Answer 20

\[ F(x) = \frac{x^3}{3} + \frac{x^2}{2} \]For us :-)

Answer 21

so i plug in the numbers to that function?

Answer 22

Yup, 9 and -a

Answer 23

is that my final answer?

Answer 24

After plugging in both numbers, yeah. Assuming that a is an arbitrary constant.

Answer 25

i got 202.5-(-a)^3/3-(-a)^2/2 im plugging it in its saying its wrong am i doing something wrong?

Answer 26

I got \[\frac{a^3}{3} - \frac{a^2}{2} + 283.5 \]Maybe check the power of nines?

Answer 27

yes that was the problem thank you so much.

Answer 28

You are welcome :-)