Question

Evaluate the integral

Answer 1

\[\int\limits_{2}^{4}(3x^{-2}-6)dx\]

Answer 2

\[\int\limits{ax^b}dx = \frac{ax^{b+1}}{b+1}\]

Answer 3

also

Answer 4

\[\int\limits_{a}^{b}f(x) dx=\int\limits_{}^{}f(b)dx-\int\limits_{}^{}f(a)dx\]

Answer 5

^ is fundamental theorem of calculus.

Answer 6

that's not the fundamental theorem of calculus

Answer 7

No, I typed it wrong, didn't I?

Answer 8

the fundamental theorem of calculus is\[\frac d{dx}\int_a^b f(t)dt=f(b)b'-f(a)a'\]

Answer 9

it's "fundamental" because it relates the idea of the derivative and integral as inverses of each other

Answer 10

so what is the answer I got -3/4 but I am not sure that its correct.

Answer 11

I guess I shoud have made that partial wrt x on a and b above to cover all bases, but whatever

Answer 12

\[\frac d{dx}\int_a^x f(t)dt=f(x)~~~~\text{where a is a constant}\]is another way it's stated, but really that's a limited case

Answer 13

@TuringTest , my 80's textbook disagrees with you, So i'm going to have to say I win this one.

Answer 14

so what is the answer I got -3/4 but I am not sure if that is correct.

Answer 15

@inkyvoyd if that is exactly what you book says then it is wrong.

Answer 16

@Zarkon , I doubt my book actually says this, but I like to disagree just to disagree.

Answer 17

that is part of the corollary statement if I remember correctly
and fyi, disagreeing with Zarkon is usually a bad idea

Answer 18

It says, there are two fundamental theorems of integral calculus

Answer 19

I think inky's right.

Answer 20

The first is what I said.

Answer 21

The second is what you said.

Answer 22

even my textbook agrees with inky's

Answer 23

\[\int_{a}^{b}f(x)dx=F(b)-F(a)\]

Answer 24

The corollary of your theorem, the second one, is that if y=f(x) is contiuous, there exists a integral blah blah blah

Answer 25

where \[F'(x)=f(x)\]

Answer 26

ohh yeah, Zarkon's right again. :facepalm:

Answer 27

@Zarkon , my bad. There's a difference between what I said and what you said.

Answer 28

What you said is what your textbook says.

Answer 29

What I said is what my hedgehog says.

Answer 30

*what my textbook says.

Answer 31

But, my hedgehog says something else.

Answer 32

He meant antiderivative, not indefinite integral.

Answer 33

Hey, good enough. I never took trig xD

Answer 34

But my hedgehog evidently did.

Answer 35

@tburn I don't think this is correct. I got \( -\frac{45}{4}\) What did you get for the antiderivative?

Answer 36

\[-3/1x^{-1}-x\]

Answer 37

It should be: \( -\frac{3}{x} - 6x\)
Maybe check your exponentiation?

Answer 38

Okay, thanks