Question

Integral, by the way x and y are independent variables.

Answer 1

\[\frac{d}{dx}\int_a^x f(x,y)dy\]

Answer 2

Medal if you can show your steps to evaluate this without evaluating the integral:

\[\frac{d}{dx} \int_0^x \cos(xy)dy\]

Answer 3

Differentiation under integral sign

Answer 4

Careful!

\[\frac{d}{dx}\int_a^x f(x,y) dy \ne \int_a^x\frac{\partial}{\partial x} f(x,y)dy\]

Answer 5

\(\Large =\cos x^2 + \int y (-\sin (xy))dy\)
Hmm.. i initially thought the 2nd integral would be easy to solve

Answer 6

Yeah just having fun idk bored lol

Answer 7

\(\Large I=\cos x^2 + \int \limits_a^xy (-\sin (xy))dy \\ \Large I = \cos x^2 -xsinx^2 +\int \limits_a^x y(-\cos (xy)\times y dy \\ \Large I = \cos x^2 - x \sin x^2 -y^2 I \)

now just isolate I :)

Answer 8

turns out to be
\(\Large I =\dfrac{\cos x^2 -x \sin x^2 }{1+y^2}\)

Answer 9

... I isn't the integral

Answer 10

Haha this is interesting and tricky I don't know, maybe something to do with that y, since the final integral should only depend on x.

Answer 11

oops, more errors.. ok
\[-2\cos(x^2)x\]

And if not, why can't we use the Second Fund. Theorem of Calc.

Answer 12

the trivial way is easy,

calculate integral, then calculate derivative.

derivative of \(\dfrac{\sin (x^2)}{x} = 2 \cos(x^2)-\dfrac{\sin(x^2)}{x^2}\)
which is the answer.

Answer 13

So \[\int_0^x f(x,y) dx \ne F(x,x)\]

Answer 14

?

Answer 15

F(x,x) - F(x,0)

Answer 16

oh ya.. silly. mixing FtoC 1 and 2. In this Q though \[\frac{d}{dx}\int_0^x f(x,y) dx = \frac{d}{dx} F(x,x) = f(x,x)?\]

Answer 17

so basically cos(x*0)---> cos(x^2)

Answer 18

getting infintessimally small rectangles for each number from 0 to x
then we gonna differentiate that wrt to x

Answer 19

no Red , F(x,0) is a function of x, so while taking the derivative w.r.t x, it cannot be ignored.

Answer 20

so lets just see how this function changes as a function of x for each cos(x*0),cos(x*1), cos(x*2),...., cos(x*x)

Answer 21

and thats always gonna be -sin(x) * y

Answer 22

i mean -sin(xy)*y

Answer 23

hmm doesnt satisfy that the derivative shud be -2sin(xy)*y, when y=x though

Answer 24

oh i know!!

Answer 25

we wwill always say y is some ratio of x

Answer 26

instead of doing the it y=0 to x
we can make sure it still hits all the numbers
cos(x*0)--->cos(x^2)
by doing ummm

Answer 27

cos(x*x/y)
setting y from y = infinity to 1