Integrate $$\int \frac{x \ \cos^{-1} x}{\sqrt{(1-x^2)}}$$

Question

diyadiya · Answer

By parts

diyadiya · Answer

I took Cos^-1x as 1st function & x/sqrt(1-x^2) as 2nd

myininaya · Answer

$$y=\cos^{-1}(x) => \cos(y)=x => -\sin(y) dy= dx$$

Recall cos(y)=x/1  (=adj/hyp)

pretend we drew a right triangle labeling the adj side to angle y x
the hyp 1 and we use the Pythagorean thm to find the opposite side to angle y

$$- \sqrt{1-x^2} dy =dx => dy=\frac{dx}{-\sqrt{1-x^2}}$$
So we have
$$-\int\limits_{}^{}\cos(y) \cdot  y dy$$
I like this form to integrate more...

myininaya · Answer

$$-\sin(y) \cdot y - \int\limits_{}^{} -\sin(y) dy=-\sin(y) \cdot y-\cos(y)+C$$

diyadiya · Answer

Hm okay But the answer is different

myininaya · Answer

$$- \frac{\sqrt{1-x^2}}{1} \cdot \cos^{-1}(x)-x+C$$

myininaya · Answer

Using the right triangle I was able to put it back in terms of x

diyadiya · Answer

Lol yeah ,Thats the answer

diyadiya · Answer

Can you show me another method ? If you dont mind , this is confusing me a bit

diyadiya · Answer

Its okay wait , i'll just go through this again

diyadiya · Answer

$$- \sqrt{1-x^2} dy =dx  $$ How did you get this ?

myininaya · Answer

The part where I drew a right triangle?

myininaya · Answer

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