Question

proof integration
sec^2 =tan x

Answer 1

You mean prove: \[
\int \sec^2x \;dx=\tan x+C
\]

Answer 2

yes

Answer 3

Are we allowed to use the Fundy Theo Calculus?

Answer 4

what is that??? first time i heard of it

Answer 5

Fundamental Theorem of Calculus states that an indefinite integral is just the anti-derivative

Answer 6

What are we allowed to use to prove it?

Answer 7

yes

Answer 8

Okay then, differentiate \(\tan(x)\) and you win!

Answer 9

if from differentiate from tan x i have done it...
but my lecture told me he want from integratretion of sec^2

Answer 10

Hmmm, tough!

Answer 11

\(\sec^2 x = 1/\sin^2 x\)

Answer 12

\[\sec ^{2}x= \frac{ 1 }{ \cos ^{2} }\]

Answer 13

Right... I messed up lol.

Answer 14

Wait....

Answer 15

If we can't do derivative of \(\tan\) then we cant do derivative of any other function right?

Answer 16

yes.. you right.
you only can use integration method to proof it

Answer 17

Do we have to use the definition of integral? That is too tough!

Answer 18

yes ..its very taugh...
my lecturer said no pain no gain

Answer 19

I can't really help you because I don't actually know what we are allowed to use...

Answer 20

The definition of the integral is: \[
\int\limits_a^b f(x) dx \equiv \lim_{\Delta x\to 0}\sum_if(x_i^*)\Delta x_i
\]Where the interval \([a,b]\) is partitioned into intervals, \(x_i^*\) is any element in the \(i\)th partition, \(\Delta x_i\) is the length of the \(i\)th partition, and \(\Delta x\) is the largest \(\Delta x_i\) (the largest partition length).

This definition can become less generalized by coming up with a method partitioning and choosing \(x^*\).
Consider:
- Let \(n\) be the number of partitions
- Give each partition have equal length: \(\Delta x = \frac{b-a}{n}\)
- Choose the right most element in the partition: \(x_i^*=i\Delta x\).
Notice that as \(n\to \infty,\;\Delta x \to 0\).
Our less generalized definition is: \[
\int\limits_a^b f(x) dx \equiv \lim_{n\to \infty}\sum_i^nf\left(\frac{(b-a)i}{n}\right)\frac{b-a}{n}
\]

Answer 21

i can think of this,
put t= tan x
dt = sec^2 x dx
\(\int \sec^2xdx = \int dt = t+c = \tan x+c\)
i have basically used differentiation but in the disguise of substitution. maybe it'll work :P