I am thinking a bit stupidly but here comes a thought:

Let's find the indefinite integral of $\dfrac{e^x}{x}.$

....I know it's $\text{Ei}(x)+C$ but let's do it anyways

So firstly, I'm gonna let $\dfrac{1}{x}=t,$ leading us to $x=\dfrac{1}{t}.$

Differentiating gives $dx=-\dfrac{1}{t^2}dt,$ and therefore:

$\begin{aligned} \int \frac{e^x}{x}dx &= \int \frac{1}{x}\cdot e^x dx \\ & =\int t \cdot e^{\frac{1}{t}}\cdot\left(-\dfrac{1}{t^2}\right) dt \\ & =- \int \frac{e^{\frac{1}{t}}}{t} dt \\ & =- \int \frac{e^{\frac{1}{x}}}{x} dx \\ \end{aligned}$

From above, we get:

$\displaystyle \int \frac{e^x}{x} dx + \int \frac{e^{\frac{1}{x}}}{x} dx = 0 + (C)$

$\displaystyle \int \frac{e^x+e^{\frac{1}{x}}}{x} dx = 0 + (C)$

Differentiating both sides gives:

$\displaystyle \frac{e^x + e^{\frac{1}{x}}}{x} = 0$

....then...

$\displaystyle e^x+e^{\frac{1}{x}}=0$ ...?!

I didn't expect this to happen but I have this terrible migraine to think more deeply about this ._.

Yes I am stupid, sorry ;;>_>

#Calculus

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You forgot about the arbitrary constant of integration.

Pi Han Goh - 2 years, 3 months ago

Hmm...

then let $\displaystyle \int \frac{e^x}{x}dx+\int \frac{e^{\frac{1}{x}}}{x}dx=C,$ but when you differentiate both sides,

$\dfrac{e^x+e^{\frac{1}{x}}}{x}=0.$

Boi (보이) - 2 years, 3 months ago

[ this comment is wrong ]

Pi Han Goh - 2 years, 3 months ago

@Pi Han Goh – Yes

The integral of 1 - cos^2 x is

The integral of 1 - the integral of cos^2 x

= x - the integral of cos^2 x + C

Therefore

The integral of sin^2 x + the integral of cos^2 x = x + C

Differentiate and we get

sin^2 x + cos^2 x = 1

Boi (보이) - 2 years, 3 months ago

@Boi (보이) – Awhh damn it. I'll try to fix up my reply and answer your question shortly....

Pi Han Goh - 2 years, 3 months ago

@Pi Han Goh – Arghhh, curses! I can't properly explain it. I think got something to do with inverse functions and 1-1 or something.

Summoning the great @Mark Hennings

Pi Han Goh - 2 years, 3 months ago

@Pi Han Goh – Suppose that we write $F(x) \; = \; \int_1^x \frac{e^u}{u}\,du \hspace{2cm} G(x) \; = \; \int_1^x \frac{e^{\frac{1}{u}}}{u}\,du$ Then the substitution $v = u^{-1}$ gives $F(x) \; = \; -\int_1^{x^{-1}} \frac{e^{\frac{1}{v}}}{v}\,dv \; = \; -G(x^{-1})$ and hence $F(x) + G(x^{-1}) \; = \; 0$ What you are saying is that $F(x) + G(x) \; = \; c$ which is not true. You are making the substitution $t = x^{-1}$ , and then replacing $t$ by $x$ , instead of by $x^{-1}$ . Since you are working with indefinite integrals, you cannot exchange the variables freely if they already have an interrelationship.

Mark Hennings - 2 years, 3 months ago

@Mark Hennings – AHHH AH AH AH OKE THANKS now I understand, really, thank you!

Boi (보이) - 2 years, 3 months ago

$\int \frac{e^{1/t}}{t} dt \neq \int \frac{e^{1/x}}{x} dx$ . Remember $x=\frac{1}{t}$ , and we are talking about indefinite integrals.

Abhishek Sinha - 2 years, 2 months ago

Yeah... Someone told me that in the below chain of comments and...

....I feel even dumber now. x'D

Boi (보이) - 2 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$