Writing arcsin in log form

1. Oct 11, 2007

azatkgz

It seems to that i went wrong way.
1. The problem statement, all variables and given/known data

$$\int \frac{arcsin(e^x)dx}{e^x}$$

3. The attempt at a solution
$$=-\frac{arcsin(e^x)}{e^x}+\int \frac{dx}{\sqrt{1-e^{2x}}}$$

$$\int \frac{dx}{\sqrt{1-e^{2x}}}=\frac{x}{\sqrt{1-e^{2x}}}+\int \frac{xe^{2x}dx}{(1-e^{2x})^{\frac{3}{2}}}$$

Last edited: Oct 11, 2007
2. Oct 11, 2007

ppyadof

Ever thought about writing arcsin in log form (the inverse function of sin when sin is written in complex numbers), that way something might happen between the logs and the exp's. I haven't tried it, so I don't know, its just a suggestion.

3. Oct 12, 2007

azatkgz

Here is the another way I tried.
$$u=e^{-x}$$
$$-\int arcsin(\frac{1}{u})du=-\int arccsc(u)du=-uarccsc(u)-\int\frac{du}{\sqrt{u^2-1}}$$

4. Oct 12, 2007

azatkgz

Than for
$$u=sec\theta\rightarrow du=\frac{sin\theta d\theta}{cos^2\theta}$$

Last edited: Oct 12, 2007
5. Oct 12, 2007

azatkgz

$$\int\frac{du}{\sqrt{u^2-a}}=-\int\frac{d\theta}{cos\theta}$$
But I don't know,actually,how to integrate last one.

6. Oct 12, 2007

rocomath

$$-\int\frac{d\theta}{\cos{\theta}}$$

trig identity

$$-\int\sec{\theta}d\theta$$

7. Oct 12, 2007

azatkgz

Ok,than please check everything.
$$\int \frac{arcsin(e^x)dx}{e^x}$$

$$u=e^{-x}$$

$$-\int arcsin(\frac{1}{u})du=-\int arccsc(u)du=-uarccsc(u)-\int\frac{du}{\sqrt{u^2-1}}$$

$$u=sec\theta\rightarrow du=\frac{sin\theta d\theta}{cos^2\theta}$$

$$=-uarccsc(u)+\int\sec{\theta}d\theta$$

$$=-uarccsc(u)+ln|sec\theta+tan\theta|$$

$$=-e^{-x}arccsc(e^{-x})+ln|e^{-x}+e^{-x}\sqrt{1-\frac{1}{e^{-2x}}}|$$

8. Oct 12, 2007

azatkgz

Sorry,I've posted answer to another question here.

Last edited: Oct 13, 2007
9. Oct 13, 2007

Gib Z

Post 7, you made a sign error. Suddenly the sec integral becomes positive...other than that, the answer is fine, but it would look nicer if you took the e^(-x) under the sqrt sign in the log, as the argument of the log is always positive anyway, no need for absolute value signs.

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