Question

Integrate cos(ln(x)) ...

Answer 1

\[\Huge \int\limits \cos(\ln(x))~dx\]

Answer 2

so far,
u=lnx e^u=x
du=1/x dx ------> e^u du = dx

\[\int\limits \cos(u)(e^u)~du\]
...

Answer 3

solution would be
e^2/2(cosu+sinu)

Answer 4

answer is
\[\frac{x[\cos(\ln(x))+\sin(\ln(x))]}{2}\]

Answer 5

let:
\[\int\limits e^2cosudu\]
t=cosu ds=e^udu
dt=-sinudu s=e^u
\[=e^ucosu+\int\limits e^u\sin udu=\]
p=sinu dr=e^udu
dp=cosu du r=e^u
\[=e^ucos u +e^usin u - \int\limits e^u cosu du\]
so:
\[2\int\limits e^ucos u=e^u cosu +e^u \sin u\]

Answer 6

solution would be
e^u/2(cosu+sinu)

Answer 7

\[\frac{ xsin \ln(x)+\cos \ln(x) }{ 2 }\]

Answer 8

2 IBPs, great

Answer 9

right

Answer 10

thanks

Answer 11

yw