Question

Calculate the definite x integral of 2+3x=y from x=3 to 10
Show steps please

Answer 1

First you want to calculate the antiderivative of 2+3x. Let's call that F(x). I you have that the answer will be F(10)-F(3).

Do you know how to calculate F?

Answer 2

No asume I know nothing because I have confused myself

Answer 3

The idea of the antiderivative is the following: if you derive it you'll get back at the original function. So dF(x)/dx=2+3x

You compute the antiderivative of a polynomial as follows: add one to the degree of every term and divide by the degree +1.

Some examples: antiderivative of x: degree is one, add one to the degree to get 2 and divide by 2: we get 1/2 x^2

antiderivative of 2X^3+6: for the first term we get 2/4 x^4, the degree of the the second term is 0, so we get 6/1 x. all in all we get 1/2 x^4+6x

Answer 4

Is it clear so far?

Answer 5

I think so keep going

Answer 6

If you're with me so far, can you compute the antiderivative of 2+3x?

Answer 7

\[1+(3/2)^2\]

Answer 8

what's the derivative of that?

Answer 9

is that correct before I do

Answer 10

nope, sorry, you'll see when you compute the derivative.

Answer 11

I got 0 as the derivative of my antiderivative

Answer 12

Right, but it should be 2+3x, so it's not correct.

Answer 13

ok

Answer 14

You left out the x's in your antiderivative. It should be 2/1 x + 3/2 x^2. which simplifies to 3/2 x^2+2x

Clear?

Answer 15

is it \[(((3x)^2)/2) + 2x\]

Answer 16

oh i see your now

Answer 17

what is the next step

Answer 18

So the 3 isn't squared. We have F(x)=3/2 x^2 +2x.

Now you need to compute F(10)-F(3)

Answer 19

so just plug 3 in and 10 in and get two answeres and then subtract the answer from 3 from 10

Answer 20

that's right.

Answer 21

ok let me works it

Answer 22

I got f(10) = 170 and f(3) = 33/2

Answer 23

is this correct?

Answer 24

F(3) is something else I believe.

Answer 25

got 39/2.... sorry... so final answer 150.5 or 301/2

Answer 26

Thanks so much!!!

Answer 27

could you help me with another

Answer 28

Great, Also, the uppercase ia important in the notation: F(x) denotes the antiderivative of f(x)

Answer 29

ok

Answer 30

\[\int\limits_{?}^{?} 2e^{5x} dx\]

Answer 31

all it says is to solve

Answer 32

When there are no boundaries in you integral you have an indefinite Integral. So earlier we had the boundaries 3 and 10, that was a definite integral, now we have an indefinite one. This means that we don't have to the second step of computing F(upper bound)-F(lower bound), we just need to compute the antiderivative of 2e^(5x)

Answer 33

ok I will try

Answer 34

When you derive a power of e you multiply by the constant in front of the x right? So d/dx e^(4x)=4e^(4x).

The derivative of our antiderivative should 2e^(5x). So we need to compensate for the 5 you get when deriving the antiderivative.

Answer 35

meaning?

Answer 36

2e^5 x\[2e^5 x\]

Answer 37

It's 2e^(5x), right?

Answer 38

yes you are right

Answer 39

The antiderivative here is the original function divided by 5. So F(x)=2/5 e^(5x). (*)
So you can compute the derivative of this, see if it's correct.

* this isn't completely true, we'll talk about that later.

Answer 40

so the antiderivative is \[(2e ^{5x})/5\]

Answer 41

right, but let's talk about (*). What's the derivative of 2/5 e^(5x) +1?

Answer 42

it is the original 2e^(5x)

Answer 43

So apperently 2/5 e^(5x)+1 is also an antiderivative.

Answer 44

ok so what do i need to do with that information

Answer 45

You can add any constant to the antiderivative and still get an antiderivative, so we denote this by adding +C, to the answer. So the actual solution to the integral here is 2/5 e^(5x)+C.

You always need to add that C if you compute indefinite integrals, it's not important in definite integral.

Answer 46

oh ok I see. that was explained in the book but no clear at all
Thanks again you were a huge help

Answer 47

great, you're welcome.