Question

Calculus help integrals

Answer 1

If f is an antiderivative of \[\frac{ \sin ^{2}x }{ x ^{2}+2 }\]
such that f(2)=1/2 then f(0)=?

Answer 2

looks like no closed form
http://www.wolframalpha.com/input/?i=integrate+sin%28x%29%5E2%2F%281%2Bx%5E2%29

Answer 3

What does that mean?

Answer 4

answer is not nice ..

Answer 5

i think maybe this is supposed to be a numerical exercise

Answer 6

it has to be

Answer 7

the idea if i am correct, is that
\[\int_a^2f(t)dt=\int _a^0f(t)dt+\int_0^2f(t)dt\] or
\[\int_a^0f(t)dt=\int_a^2f(t)dt-\int_0^2f(t)dt\]

Answer 8

according to Mathematica, answer is
0.0306445

Answer 9

you are told that
\[\int_a^2f(t)dt=\frac{1}{2}\] and so your job is, using a computer algebra system, to find
\[\int_0^2f(t)dt\] and subtract that result from \(\frac{1}{2}\)

Answer 10

0.0306445 is not one of my choices but also make sure you entered it right in wolfram you had x^2+1 and it is x^2+2

Answer 11

@experimentX you can't know directly from mathematica what that integral is, because you don't know the lower limit of integration

Answer 12

woops!! 0.174801
just find the value of constant ...

Answer 13

subtract this number from \(.5\) and see if you get one of the answer choices
http://www.wolframalpha.com/input/?i=integral+0+to+2+%28sin^2%28x%29%29%2F%28x^2%2B2%29

Answer 14

0.175 is one of the options so that sounds right

Answer 15

http://www.wolframalpha.com/input/?i=.5+-+integral+0+to+2+%28sin^2%28x%29%29%2F%28x^2%2B2%29

Answer 16

-0.325 is an option but not 0.325

Answer 17

look at second link, i think you will have it

Answer 18

alright I have two more if you can stay and help

Answer 19

post new question lol ... and solve the other part numerically.

Answer 20

the graph of f prime is the line shown if f(0)=2.5 then f(2)=?

Answer 21

here i found the equation since I can see that it is linear with a slop of -4 it is y'=-4x so you integrate that and get -2x^2+C

Answer 22

your f' is linear ... your f must be quadratic. just find the solve from the graph.

Answer 23

yes and so f(0)=2.5 so if I plug that in I get C=2.5 but that causes a problem because none of the options are negative and f(2) would be negative

Answer 24

f(0) = -0^2+C = 2.5

Answer 25

that line looks like \(y=-4x+8\) to me

Answer 26

you're right that's what I was missing thank you I can take it from there

Answer 27

making your anti derivative
\[f(x)=-2x^2+8x+c\] solve for \(c\)

Answer 28

ok the last one
If f is continuous for all real numbers, \[\frac{ dy }{ dx }=f(x)\] and y(-1)=6 then y(x)=?

Answer 29

is y = f(x) ??

Answer 30

cant wait to see this...

Answer 31

seems like not enough information to me

Answer 32

I dont know that is all the question says Im completely stumped on this

Answer 33

is f(x) any arbitrary function or some information given on f(x).

Answer 34

what are your options?

Answer 35

the answers are \[A) 6+\int\limits_{-1}^{x}f(t)dt\]\[B) f(x)-f(-1)\]\[C)\int\limits_{-1}^{x}f(t)dt-6\]\[D)\int\limits_{-1}^{x}f'(t)dt\] E) none of these

Answer 36

\[ y(x) = \int_0^x f(t) dt + C \\
\int_0^{-1} f(t) dt + C = 6 \\
y(x) = \int_0^x f(t) dt + 6 - \int_0^{-1} f(t) dt = \int_{-1}^x f(t) dt + 6\]

Answer 37

What does the box lower bound mean

Answer 38

huh?

Answer 39

the first line you use the integral from that symbol to x and satellite used a similar symbol on the other problem what does it represent and why is it there?

Answer 40

BRB ... 5 min

Answer 41

typically you integrate like this
\[ \int f(x) dx = \text{some function } g(x) \]
when you have bounds
\[ \int_a^b f(x) dx = g(b) - g(a)\]

Answer 42

when you are expected to find function of 'x' then
\[ \int_0^x f(t) dt = g(x) - g(0)\]
^^ can't use 'x' here ... use dummy variable 't' instead.

Answer 43

ok

Answer 44

thank you for all your help

Answer 45

yw