Question

DESPERATE HELP!! Evaluate exactly the value of the integral. Your work must include the use of substitution and the antiderivative.

Answer 1

Substitute \[2x ^{4}+8x\] as t

Answer 2

so what do u get if u take the derivative of that?

Answer 3

8x3+8

Answer 4

Exactly!

Answer 5

So then what?

Answer 6

The latter half of the whole expression can be simply made the derivative of first half.

Answer 7

so what is the antiderivative in total?

Answer 8

did u try solving it in the above mentioned method?

Answer 9

I'm a little confused by substitution. Can you help me through it?

Answer 10

ya sure

Answer 11

8x^3+8 can be written as (4x^3+4)*2.
So ,now can you show next step?

Answer 12

So does that make the entire anti derivative (4x^3+4)*2?

Answer 13

no u do a substitution to make ur job easier

Answer 14

This can be easily adjust in dt form if you keep the first half expression as t.

Answer 15

Adjusted*

Answer 16

\[\int\limits_{-1}^{0} t (8x^3+8)\]

Answer 17

Is that correct?

Answer 18

no it should be entirely in terms of t

Answer 19

\[2x ^{4}+8x=t\]
\[(8x ^{3}+8)dx=dt\]

Answer 20

OK that makes sense, but how do you solve using that? Where does the interval come in?

Answer 21

forget about the limits of the integral for now

Answer 22

once u solve the integral indefinitely u can always substitute the limits later

Answer 23

\[(8x ^{3}+8)dx=dt\]
\[(4x ^{3}+4)dx=\frac{ dt }{ 2 }\]

Answer 24

Or if you want to do it that way you may substitute limits like
For lower limit you can go for -4 and upper is ??

Answer 25

Not -4 actually!

Answer 26

-6

Answer 27

The lower is -1 and the upper is 0.

Answer 28

No after substituition i am talking!

Answer 29

When you put 2x^4+8x^3 as t then your lower limit changes to -6 .
Can you tell me how?

Answer 30

Is it because that piece is to the third power? I really don't know.

Answer 31

Have you studied definite integration?

Answer 32

Not much.

Answer 33

did u understand till that last part i typed?

Answer 34

I understand it but I dont understand how it fits in as a whole or the next step

Answer 35

hint:
read and understand example 1(c) at the following link.
http://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleIndefinite.aspx

Answer 36

Ok that makes more sense but where did the power of three go?

Answer 37

and is the complete thing (8x^3+8)^2 + c ??

Answer 38

ALong the lines of the example,
u=2x^4+8x
du=8x^3+8=2(4x^3+4)dx
or (4x^3+4)dx=du/2
so you need to integrate
\(\large \int u^3 (du)/2\)

Answer 39

Also, it's a definite integral, so the +C does not come in.
Instead, the definite integral will be a simple number.