Solving an integral using a special substitution

**Ulysses** · June 28th 2010, 02:52 PM

Can somebody tell me if this is right?

I must solve:
$\displaystyle\int_{}^{}\displaystyle\frac{dx}{\sqr t[ ]{9+x^2}}$

And I used the substitutions:
$x=3sinht$
$dx=3coshtdt$

And the identity:

$cosh^2t-sinh^2t=1\Rightarrow{\cosht=\sqrt[ ]{1+sinh^2t}}$

$\displaystyle\int_{}^{}\displaystyle\frac{dx}{\sqr t[ ]{9+x^2}}=\displaystyle\int_{}^{}\displaystyle\frac {3coshtdt}{\sqrt[ ]{3^2+(3sinht)^2}}=\displaystyle\int_{}^{}\displays tyle\frac{3coshtdt}{\sqrt[ ]{3^2(1+sinh^2t)}}=\displaystyle\int_{}^{}\displays tyle\frac{coshtdt}{cosht}=\displaystyle\int_{}^{}d t=t+C=argsh(\displaystyle\frac{x}{3})+C$

Bye there, and thanks for posting.

**TheCoffeeMachine** · June 28th 2010, 03:46 PM

Originally Posted by Ulysses

Can somebody tell me if this is right?

Well done!

**Ulysses** · June 28th 2010, 03:59 PM

Thanks.
What about this one? I don't know how to proceed now:

$\displaystyle\int_{}^{}\displaystyle\frac{dt}{\sin (t)}$

I've solved this using the table, using the definition of cosecant: $\displaystyle\int_{}^{}\displaystyle\frac{dt}{\sin (t)}=\ln|csc(t)-cot(t)|+C$

This was the result of solving: $\displaystyle\int_{}^{}\displaystyle\frac{dx}{x\sq rt[ ]{1-x^2}}=\ln|csc(arcsin(x))-cot(arcsin(x))|+C$

But now I must solve:
$\displaystyle\int_{}^{}\displaystyle\frac{dx}{x^2\ sqrt[ ]{4-x^2}}$

$x=2\sin(t)$
$dx=2\cos(t)dt$

$\displaystyle\int_{}^{}\displaystyle\frac{dx}{x^2\ sqrt[ ]{4-x^2}}=\displaystyle\int_{}^{}\displaystyle\frac{2\ cos(t)dt}{4\sin^2(t)\sqrt[ ]{2^2-(2\sin(t)^2}}=\displaystyle\frac{1}{4}\displaystyl e\int_{}^{}\displaystyle\frac{dt}{\sin^2(t)}}=\dis playstyle\frac{1}{4}\displaystyle\int_{}^{}csc^2(t )dt$

I thought of solving by parts, but...

$u=csc(t)$
$du=-csc(t)cot(t)dt$

$dv=csc(t)dt$
$v=\ln|csc(t)-cot(t)|$

**TheCoffeeMachine** · June 28th 2010, 04:30 PM

Originally Posted by Ulysses

What about this one? I don't know how to proceed now:

$\displaystyle\int_{}^{}\displaystyle\frac{dt}{\sin (t)}$

Do you know the Weierstrass substitution? If If [LaTeX ERROR: Convert failed] , then [LaTeX ERROR: Convert failed] and [LaTeX ERROR: Convert failed] . Using these results we have [LaTeX ERROR: Convert failed]

But now I must solve:
$\displaystyle\int_{}^{}\displaystyle\frac{dx}{x^2\ sqrt[ ]{4-x^2}}$

$x=2\sin(t)$
$dx=2\cos(t)dt$

$\displaystyle\int_{}^{}\displaystyle\frac{dx}{x^2\ sqrt[ ]{4-x^2}}=\displaystyle\int_{}^{}\displaystyle\frac{2\ cos(t)dt}{4\sin^2(t)\sqrt[ ]{2^2-(2\sin(t)^2}}=\displaystyle\frac{1}{4}\displaystyl e\int_{}^{}\displaystyle\frac{dt}{\sin^2(t)}}=\dis playstyle\frac{1}{4}\displaystyle\int_{}^{}csc^2(t )dt$

[LaTeX ERROR: Convert failed] is a standard integral. As long you broke down the integral up-to this stage and know this result you should be alright.

**Ulysses** · June 28th 2010, 04:39 PM

Thank you verymuch!

**TheCoffeeMachine** · June 28th 2010, 04:42 PM

Originally Posted by Ulysses

Thank you verymuch!

No problem. I take a delight in helping people who show their work, if I can that is.

**Ulysses** · June 28th 2010, 05:05 PM

Really noble from you sharing your knowledge that way. Believe me, I know that people like you makes this world a better place. I know... a post in a forum doesn't mean much, but is the attitude. And actually the free access to knowledge makes the difference.

Thanks is all I can give now

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