derivative of an integral!!! function is ln(1+t^2) and the Intergal from x=1 to x

Question

derivative of an integral!!! function is ln(1+t^2) and the  Intergal from x=1 to x

across · Answer

This thing here makes no sense. I suppose you're trying to evaluate$$\int_{1}^{x}\ln(1+t^2)\text{ dt?}$$

anonymous · Answer

$$  y=\int\limits_{1}^{x}\ln(1+t ^{2}) dt$$

anonymous · Answer

question says find dy/dx of that integral

across · Answer

FTC

anonymous · Answer

whats FTC?

turingtest · Answer

the fundamental theorem of calculus

anonymous · Answer

oh. i've never heard of it :/

across · Answer

That's like doing algebra and not knowing what multiplication is. ;P

Give it a look.

amistre64 · Answer

your asking a calculus question and have never heard of the fundamental thrm of calculus?

turingtest · Answer

basically, if$$F(x)=\int_{a}^{x}f(t)dt$$then$$F'(x)=f(x)$$

anonymous · Answer

oh ok yeah the derivative of an antiderivative is = f(x)

amistre64 · Answer

i think turings notation is a little off

turingtest · Answer

the derivative is the inverse of the antiderivative (integral)
a rather fundamental idea, hence the name

what did I do amistre?

anonymous · Answer

it's right. his explanation is the second fundamental! I remember now

turingtest · Answer

amistre64 · Answer

$$F(t)=\int f(t)dt$$
$$\int_{a}^{b} f(t)dt=F(b)-F(a)$$
$$\frac{d}{dx}F(b)-F(a)\to\ F(b)b'-F(a)a'$$

anonymous · Answer

it's not a to b though. it's a to x

amistre64 · Answer

sigh ... a and b are generic place holders

anonymous · Answer

but there's a theorem for a to x

turingtest · Answer

that's what I know as the second part.. sort of
don't recognize the a' and b' part though
I guess I always assume a and b are constant

turingtest · Answer

...but mine is right too!

across · Answer

Amistre just generalized it.

anonymous · Answer

we learnt about two fundamentals 1) is the one u stated and 2) is the one turing stated

amistre64 · Answer

:)

turingtest · Answer

yeah, that is more general... you win this round amistre XD

amistre64 · Answer

but i see an error in mine; I F-ed on the derivaive instead of f-ing it lol

across · Answer

That sounds so wrong.

amistre64 · Answer

im getting cramps im laughing so hard from that one

anonymous · Answer

:/

turingtest · Answer

still stuck ?

amistre64 · Answer

the point being; we know f, a and b so it boils down to plugging it in

f(b)b' - f(a)a'

anonymous · Answer

ok.

amistre64 · Answer

f(x)x' - f(1)1' = f(x) = ln(1+x^2)

anonymous · Answer

my prof said the answer was ln(1+x^2)2x-ln(1+x^4)4x^3

anonymous · Answer

ok. i solved it. never mind. Thank you for your help