Question

how to integrate cos (hat)*cosat??

Answer 1

What?

Answer 2

it's cos hyperbolic at multiplied by cos at

Answer 3

\[\int\limits \cosh(x)\cos(at)dx?\]

Answer 4

use IE by parts ... twice

Answer 5

If that's your integral cos(at) is just a constant...

Answer 6

\[\int\limits_{0}^{t} \cosh at \cos at dt\]

Answer 7

Oh, then use what @experimentX said

Answer 8

can you please explain how to use that

Answer 9

\[\huge \int\limits u dv = uv - \int\limits v du\]

Int. by parts

\[\huge u = \cos(at) ~~ du = - (a*\sin(at))dt \]

\[\huge dv = \cosh(at) dt ~~ v = \frac{ \sinh(at) }{ a }\]

Answer 10

But you have to do this twice

Answer 11

or you can alternatively try changing into polar form.

Answer 12

\[\int\limits \cos(at)\cosh(at)dt =~ \frac{ \cos(at)\sinh(at) }{ a }+\int\limits \sin(at)\sinh(at)dt\]

Now use by parts again.

Answer 13

ok ...thanks for your help

Answer 14

Np :)

Answer 15

You could take the real/imaginary part of a complex integral possibly.

Answer 16

∫cos(at)cosh(at)dt= cos(at)sinh(at)a+∫sin(at)sinh(at)dt
\[=\cos(at)\sinh(at)\div a- \sin(at) \cos h(at)\div a +\int\limits \cos (at) \cosh(at)dt\]

Answer 17

now this cancels out the left hand side integral

Answer 18

\[\cosh(at)=\frac{e^{at}+e^{-at}}{2} \]\[\cos(at)=\frac{e^{iat}+e^{-iat}}{2} \]