How do you solve this calculus problem?

**honestliar** · August 15th 2009, 08:13 PM

$\int x(1-2e^{cotx^{2}})csc^{2}(x^{2})dx$

$\int5 e^{\frac{1}{2}lnx}dx$

$\int ln e^{\frac{x}{2}}dx$

Can you show how it's done?

Thank you

**songoku** · August 15th 2009, 08:35 PM

Hi honestliar

Do you mean integration ?

1. let : $u=1-2e^{(\cot x^2)}$

2. hint : $e^{\ln a}=a$

3. hint : $\log a^b = b*\log a$

**tom@ballooncalculus** · August 17th 2009, 05:01 AM

Originally Posted by honestliar

$\int x(1-2e^{cotx^{2}})csc^{2}(x^{2})dx$
[snip]

Expand the brackets and tackle 2 separate integrals.

Just in case a picture helps...

As usual, straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed lines similarly but with respect to the dashed balloon expression. So the triangular network...

... is the chain rule.

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**songoku** · August 17th 2009, 09:10 PM

Hi honestliar

1.
let : $u=1-2e^{(\cot x^2)}$

$du=-2e^{(\cot x^2)}\; (-\csc^{2}(x^{2})) \; (2x) \; dx= 4x\; e^{(\cot x^2)}\; \csc^{2}(x^{2})\; dx$

Can you continue ?

P.S: I just realized that Mr. F edited my post. Thx Mr. F

**tom@ballooncalculus** · August 18th 2009, 01:07 AM

Hi, songoku - I couldn't (yet) make your sub (with parts, maybe?) work for the big one. Hope mine was ok...

**songoku** · August 18th 2009, 05:07 AM

Hi tom@ballooncalculus

Your method is ok. Hope honestliar understands it

As for mine, subs. dx into the question and state $e^{(\cot x^2)}$ in term of u.

**tom@ballooncalculus** · August 18th 2009, 02:01 PM

D'oh! Cheers for that... and if you don't mind another picture...

... where we're doing just as you say, expressing $e^{\cot(x^2)}$ in terms of u (i.e. the dashed balloon expression)...

... in order to divide by $e^{\cot(x^2)}$ and thus cancel the extra one appearing in the derivative by-product. I'm using F and G just to point up how the integration with respect to u splits into two.

Obviously, the lower equals sign depends on several lines of algebra that aren't shown, so the picture's only an overview.

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