Question

Could I get a quick double-check on this integral?

Answer 1

\[\int e^{\sin x}({x\cos^3x-\sin x\over\cos^2x})dx=\int e^{\sin x}x\cos xdx+\int e^{\sin x}({-\sin x\over\cos^2x})dx\]integrate each one by parts individually
the first integral:\[u=x;dv=e^{\sin x}\cos x\]\[\int e^{\sin x}x\cos xdx=xe^{\sin x}-\int e^{\sin x}dx\]for the second integral\[u=e^{\sin x};dv={-\sin x\over\cos^2x}\]\[\int e^{\sin x}({-\sin x\over\cos^2x})dx=-e^{\sin x}\sec x+\int e^{\sin x}dx\]add the two integrals together and you see that \(\int e^{\sin x}dx\) cancels, so we get\[\int e^{\sin x}({x\cos^3x-\sin x\over\cos^2x})dx=\int e^{\sin x}x\cos xdx+\int e^{\sin x}({-\sin x\over\cos^2x})dx\]\[=xe^{\sin x}-e^{\sin x}\sec x+C=e^{\sin x}(x-\sec x)+C\]

Answer 2

:)

Answer 3

can you use wolframalpha to check it?

Answer 4

I have not been trusting the wolf recently... it gives the answer but not the process, so I am more concerned with seeing if I have committed an error in the logic of the process
plus I wanna see a better way to do it if possible

Answer 5

wolf has been wrong on many occasions as well

Answer 6

wolf can be wrong when input is entered wrong. the chance that wolf is wrong inside its engine is nearly zero.

Answer 7

good to know. I just discovered wolf and I use it as a double check. I hope someone can help you find an easier way to do the problem.
Integration by parts looks like the easiest way to do this one.

Answer 8

wolf is a program. Programs can be wrong that is why they have to be debugged.

Answer 9

@GT I can provode you with a few of wolf's mistakes
it messes up a lot with DE's and FoolForMath has found quite a few mistakes in other types of problems

Answer 10

your answer is correct.

Answer 11

Thanks Zarkon :D
here is an example of wolf's failure @GT
http://openstudy.com/study#/updates/4f0259eee4b01ad20b54445d

Answer 12

here is wolf giving a correct answer to an integral with the most moronic process imaginable
http://www.wolframalpha.com/input/?i=integral%28csc%28x%29%29%5E%2810%29%28cot%28x%29%29%5E3dx
click show steps and have a laugh

Answer 13

Technology has its limitations that is why it will never replace the human brain.

Answer 14

It is not the answer. The result is "computation timed out". It is too complex for the computer to solve in 10 second limit they have.

Answer 15

the answer to the link is correct, but click 'show steps'
I can do that problem in 3 or 4 steps
wolf did it in like 150 lol

Answer 16

How about overflow on the graphing calculator?
\[\frac{9^\left( 2001 \right)}{9^\left( 2000 \right)}\]

Answer 17

I don't even use graphing calculators very much
but that problem is trivial anyway...

Answer 18

Same concept, technology has its limitations

Answer 19

indeed :)

Answer 20

my calculator does that with no problems

Answer 21

wolf should too I think...

Answer 22

any caclulator with a cas will find it easy

Answer 23

but certainly the other examples I provided are sufficient to demonstrate wolfs limits
plus, wolf does things like ALWAYS use l'hospital for limits, ALWAYS use reduction formulas for trig integrals; things that are completely illogical from a human thought standpoint

Answer 24

yes...wolf definitely has its limitations...I have seen other mistakes it has made. Use Mathematica if you have it available.

Answer 25

yeah, as soon as I feel grown-up enough ;)