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Dido525
 2 years ago
Evaluate the Intergal
Dido525
 2 years ago
Evaluate the Intergal

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Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0\[\int\limits\limits_{}^{}\frac{ \arcsin(x) }{ x^2 } dx\]

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0I have no idea... I am think about trig substitutions but I am not sure about this...

wio
 2 years ago
Best ResponseYou've already chosen the best response.1Not sure either, but what does \(x=\sin(u)\) get you?

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.0I was thinking about trying integration by parts... have you tried that yet? :)

wio
 2 years ago
Best ResponseYou've already chosen the best response.1It would get rid of the arcsin.

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Well I was going to make a substitution and then use intergration by parts but I am not sure...

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0What if I said u=arcsin(x) ?

wio
 2 years ago
Best ResponseYou've already chosen the best response.1That's the same as x = sin(u)

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Right. But if I said that I would get a squart root and I could use trigonometric substitution.

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Allright so I so far got... Use intragtion by parts: u=arcsinx) \[du=\frac{ 1 }{ \sqrt{1x^2} }\] dv=x^2 dx v=x^1

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0\[\frac{ \arcsin(x) }{ x }\int\limits_{}^{}\frac{ 1 }{ x \sqrt{1x^2} }dx\]

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0I am kinda stuck at this point.

BAdhi
 2 years ago
Best ResponseYou've already chosen the best response.2First do the substitution $$x=\sin(u)\implies dx=\cos(u)du$$ then $$\int \frac{\arcsin{x}}{x^2}\;dx=\int \frac{u\cos(u)}{\sin^2(u)}\;du=\int u\csc(u)\cot(u)\;du$$ $$\int u\frac{d(\csc(u))}{du}du$$ do the integration by parts

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0@BAdhi So I got to this point \[\int\limits_{}^{}ucsc(u)\cot(u) du\]

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0I would use integration by parts here right?

BAdhi
 2 years ago
Best ResponseYou've already chosen the best response.2we know that $$\frac{d\csc(u)}{du}=\csc(u)\cot(u)$$ then, $$\begin{align*}\int u\csc(u)\cot(u)\,du&=\int u\frac{d\csc(u)}{du}\,du\\ &=u\csc(u)+\int \csc(u)\,du \end{align*}$$

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Now I need to figure out the integral of csc(x).
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