Question

Find the integral of 1/(sqrt(t^3+2)) dt from 0 to x.

Answer 1

\[\int_0^x\frac{1}{\sqrt{t^3+2}}dt\]?

Answer 2

you are not going to find a nice closed form for this
what is the actual question?

Answer 3

Yes. Please continue.

Answer 4

what does the question say?
my guess: "find the derivative" but i could be wrong

Answer 5

it cannot really say "find the integral" because it is an integral

Answer 6

It's a multiple choice question. If f(x)=integral of 1/(sqrt(t^3+2)) dt from 0 to x, which of the following is FALSE? a) f(0)=0 b) f is continuous at x for all x greater than or equal to 0. c) f(1)>0 d) f'(1)=1/sqrt(3) e) f(-1)>0

Answer 7

ok now we have a question

Answer 8

\[f(x)=\int_0^x\frac{1}{\sqrt{t^3+2}}dt\]
\[f(0)=\int_0^0\frac{1}{\sqrt{t^3+2}}dt=0\]

Answer 9

Why does f(0)=0?

Answer 10

so first one is right
also \(f(x)=\int_0^x\frac{1}{\sqrt{t^3+2}}dt\) is a continuous function, so second one is right

Answer 11

\[f(0)=\int_0^0\frac{1}{\sqrt{t^3+2}}dt\] yes?

Answer 12

you are integrating over no path, or a path with length zero, so the integral is 0

Answer 13

Okay, continue.

Answer 14

\[f(x)=\int_0^x\frac{1}{\sqrt{t^3+2}}dt\]
\[f(1)=\int_0^1\frac{1}{\sqrt{t^3+2}}dt\] and this is positive because the integrand is positive on this interval

Answer 15

And how is f(1) positive?

Answer 16

because the integrand is positive on the interval \((0,1)\)

Answer 17

You mean 1/(sqrt(t^3+2)) is positive on (0, 1)?

Answer 18

yes

Answer 19

Okay, continue.

Answer 20

\[f(x)=\int_0^x\frac{1}{\sqrt{t^3+2}}dt\]
\[f'(x)=\frac{1}{\sqrt{x^3+2}}\] and so
\[f'(1)=\frac{1}{\sqrt{3}}\]

Answer 21

Okay, continue?

Answer 22

i guess the last one has to be wrong

Answer 23

since the intgrand is positive on \((-1,0)\) then
\[f(-1)=\int_0^{-1}\frac{1}{\sqrt{t^3+2}}dt\] must be negative

Answer 24

because you are going backwards, from \(0\) to \(-1\)
this
\[\int_{-1}^0\frac{1}{\sqrt{t^3+2}}dt\] would be positive

Answer 25

Thank you.

Answer 26

yw