Question

Any reason why i cant rewrite this integral as this?

Answer 1

\[\int\limits \frac{ COS(\sqrt{X} }{ \sqrt{X} } = \int\limits COS (x ^{1/2}) * x ^{-1/2}\]

Answer 2

thats fine

Answer 3

Then i can just use by parts to solve it.

Answer 4

you can always give it a shot

Answer 5

or we can assume that this MIGHT come from something similar to sin(x^(.5))

\[D_x[sin(x^{.5})]=\frac{cos(x^{.5})}{2x^{.5}}\]

notice that if we introduce a constant 2 at the onset, we get what we are looking for

\[D_x[2sin(x^{.5})]=\frac{2~cos(x^{.5})}{2~x^{.5}}\]

Answer 6

and of course any general "+C" tacked on gives us a family of results

Answer 7

im seeing it

Answer 8

u-sub would amount to:\[u=x^{1/2}\]\[du=\frac{1}{2x^{1/2}}dx\]\[2du=\frac{1}{x^{1/2}}dx\]and by substituting
\[\int\frac{cos(x^{1/2})}{x^{1/2}}dx=>\int cos(u)\frac{1}{x^{1/2}}dx=>\int cos(u)2du\]

therefore

\[\int 2cos(u)~du\]

Answer 9

\[\int\limits COS(X ^{1/2)}) * \frac{ 1 }{ x ^{1/2} }\]

Answer 10

\[-2\sin \sqrt{x} + C\]