Hi, you might have come across:\(\displaystyle \int e^{ax}\cos bx\) dx,

How do you solve it? You might use integration by parts, and also complex numbers, and I find use of complex numbers interesting!

Say \(A = \displaystyle \int e^{ax}\cos bx\)dx,

and \(B = \displaystyle \int e^{ax}\sin bx\) dx

Hence, \(A + iB = \displaystyle \int e^{ax} (\cos bx + i \sin bx)\)dx = \(\displaystyle \int e^{ax} (e^{i bx})\)dx

= \( \displaystyle \int e^{(a+ib)x}\)dx

= \( \displaystyle \frac{e^{(a+ib)x}}{a+ib}\)

= \(\displaystyle \frac{e^{(a+ib)x}(a - ib)}{a^2 + b^2}\)

\( \Rightarrow A + iB = z = \displaystyle \frac{e^{ax} (\cos bx + i \sin bx)(a - ib)}{a^2+b^2}\)

Clearly,

\(A = \text{Re}(z) = \displaystyle \frac{e^{ax}}{a^2+b^2} (a \cos bx + b \sin bx)\)

\(B = \text{Im}(z) = \displaystyle \frac{e^{ax}}{a^2+b^2} (a \sin bx - b \cos bx)\)

You can try to find this definite integral:

**Problem:** \(\displaystyle \int_{0}^{\pi} e^{(\cos x)} \cos(\sin x)\) dx

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestShort and Sweet! :)

Log in to reply

Thanks!

Log in to reply

\(cos { (\sin { x) } } =\quad ({ e }^{ isinx }+{ e }^{ -isinx })/2\\ { e }^{ cosx }\cos { (\sin { x) } } =\quad ({ e }^{ cosx+isinx }+{ e }^{ cosx-isinx })/2=({ e }^{ { e }^{ ix } }+{ e }^{ { e }^{ -ix } })/2\\ =((1+\frac { { e }^{ ix } }{ 1! } +\frac { { e }^{ i2x } }{ 2! } +...)+(1+\frac { { e }^{ -ix } }{ 1! } +\frac { { e }^{ -2ix } }{ 2! } +...))/2\)

In 2 more steps you will get the answer. The answer in this case is \(pi\)

Log in to reply

Solve his other question too: Problem without words !

Log in to reply

Hi Jatin! Thanks for the great post, but just one thing I was wondering that incorporating i, the complex number, and using the usual laws of Calculus, is it mathematically correct, I mean the laws of calculus is for reals. I may be fundamentally wrong somewhere but I need the answer.

Log in to reply

Awesome article.

By using your method I arrive at a step from where I can't proceed further.

The step is: \(A+iB=\int (e)^{e^{ix}}\,dx\), where \(A=\int (e)^{(cos x)}cos(sin x)\,dx\) and \(B=\int (e)^{(cos x)}sin(sin x)\,dx\).

How to do after this?Precisely,what's the real part?

Log in to reply

Hi, Bhargav, you are close, try to use expansion for \(e^x\).

Log in to reply

Is the answer \(\pi\)?

Log in to reply

Log in to reply

\(W0w\) \(£xcellent approach\)

Log in to reply

It is an amazing approach to solve such problems.

Log in to reply

THANKS. VERY POWERFULL METHOD-CAN BE TACKLED ANY COMPLICATED INTEGRALS LIKE THIS ONE.

Log in to reply

Finally I got it! :)

Log in to reply

Interesting method!

Log in to reply

Easy approach....nice trick...

Log in to reply

Thanks a ton..!

Log in to reply

Answer is π

Log in to reply

Too beautiful

Log in to reply

great..:-)

Log in to reply

-e^{-1} -1 is the ans

Log in to reply

Awesome!!

Log in to reply

You came our with a simple and an interesting solution the technique that triggered me was to solve it using By Parts Method but have to admit your approach was far far better than mines Can you tell me how do you get such ideas at and tender age of 15(Just Asking)?

Log in to reply

Hi, solving this integral using complex numbers is well(not very much well though) known. I did not come up with it myself.

Log in to reply

Yeah, actually the same technique's been discussed with us at our insti as well... anyways, it's really good...

Log in to reply