Question

Please help I will medal
why does d/dx S from 0 to pi cosx^2 dx=0

Answer 1

\[\frac{ d }{ dx }\int\limits_{0}^{\pi}cosx^2dx = 0\]

Answer 2

why does it equal zero???

Answer 3

since the integral is a real number

Answer 4

and not a function

Answer 5

i still don't get it

Answer 6

what does it mean when d/dx is on the outside versus being on the inside of the integral?

Answer 7

we can write this:
\[\frac{d}{{dx}}\int_0^\pi {{{\left( {\cos x} \right)}^2}} dx = \frac{d}{{dx}}\left. {\left( {\frac{{x + \sin x\cos x}}{2}} \right)} \right|_0^\pi = \frac{d}{{dx}}\left( {\frac{\pi }{2}} \right) = 0\]

Answer 8

how did you get the stuff inside the parenthesis?

Answer 9

I have used this identity:
\[{\left( {\cos x} \right)^2} = \frac{{1 + \cos \left( {2x} \right)}}{2}\]

Answer 10

If we pretend that d/dx isn't there, would you agree that definite integral evaluates to some real number ?

Answer 11

yes

Answer 12

whats the derivative of a real number ?

Answer 13

0

Answer 14

so what if the d/dx is on the inside?

Answer 15

it isn't on the inside

Answer 16

it is outside to make your life easy :)

Answer 17

if it is inside, you will have to find the derivative first and then evaluate the integral