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Integrate (cos^-1x)^2 /(1-x^2)^1/2

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Integrate (cos^-1x)^2 /(1-x^2)^1/2

Guest Jun 19, 2015

#2
+90968
+13

Thanks Syllogist.  You're right of course, I just want to try and explain.

$$\\\int\;\frac{(cos^{-1}x)^2 }{(1-x^2)^{1/2}}\;dx\\\\ What I notice is that \;\; \frac{d}{dx}cos^{-1}x=\frac{-1}{\sqrt{1-x^2}}\\\\ So I think the answer will be  some constant* (cos^{-1}x)^3 \;\;+c\\\\ Now if i differentiated this (without the constant multiple) I get\\\\ \frac{d}{dx}(cos^{-1})^3x=3(cos^{-1})^2x\times \frac{-1}{\sqrt{1-x^2}}\\\\ I need to get rid of the - and the 3 so I divide by -3 (that will be the constant multiple\\\\ \int\;\frac{(cos^{-1}x)^2 }{(1-x^2)^{1/2}}\;dx=\frac{-1}{3}cos^{-1}x+c$$

Just like Syllogist said:))

NOTE: I can never remember that inverse trig differential, I have to work it out from scratch every time.

(It doesn't take long to work out)

Melody  Jun 19, 2015
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#1
+125
+8
-1/3 cos^(-1)(x)^3 + constant
syllogist  Jun 19, 2015
#2
+90968
+13

Thanks Syllogist.  You're right of course, I just want to try and explain.

$$\\\int\;\frac{(cos^{-1}x)^2 }{(1-x^2)^{1/2}}\;dx\\\\ What I notice is that \;\; \frac{d}{dx}cos^{-1}x=\frac{-1}{\sqrt{1-x^2}}\\\\ So I think the answer will be  some constant* (cos^{-1}x)^3 \;\;+c\\\\ Now if i differentiated this (without the constant multiple) I get\\\\ \frac{d}{dx}(cos^{-1})^3x=3(cos^{-1})^2x\times \frac{-1}{\sqrt{1-x^2}}\\\\ I need to get rid of the - and the 3 so I divide by -3 (that will be the constant multiple\\\\ \int\;\frac{(cos^{-1}x)^2 }{(1-x^2)^{1/2}}\;dx=\frac{-1}{3}cos^{-1}x+c$$

Just like Syllogist said:))

NOTE: I can never remember that inverse trig differential, I have to work it out from scratch every time.

(It doesn't take long to work out)

Melody  Jun 19, 2015
#3
+18712
+10

Integrate (cos^-1x)^2 /(1-x^2)^1/2

$$\small{\text{ \begin{array}{lrcl} & \int{ \dfrac{[ \cos^{-1}{(x)} ]^2 } { (1-x^2)^{\frac{1}{2}} } \ dx } = \mathbf{ \int{ [\arccos{(x)}]^2 \cdot \dfrac{ 1 }{ \sqrt{ (1-x^2) } } \ dx } }\\\\ \end{array} }}\\\\$$

$$\small{\text{ \boxed{ \begin{array}{lrcl} \mathrm{Formula:~} & y &=& \dfrac{ f(x)^{n+1} } {n+1} \\\\ & y' &=& \left(\dfrac{n+1}{n+1}\right) \cdot f(x)^{n+1-1}\cdot f'(x) \\\\ & y' &=& f(x)^{n}\cdot f'(x) \\\\ &\mathbf{ \int{ f(x)^{n}\cdot f'(x) \ dx } } & \mathbf{=} & \mathbf{ \dfrac{ f(x)^{n+1} } {n+1} } \end{array}} }}$$

$$\small{\text{ \begin{array}{lrcl} &\mathbf{ \int{ f(x)^{n}\cdot f'(x) \ dx } } & \mathbf{=} & \mathbf{ \dfrac{ f(x)^{n+1} } {n+1} } \qquad f(x) = \arccos{(x)} \qquad f'(x)= -\dfrac{ 1 }{ \sqrt{ (1-x^2) } }\\\\ & \int{ \left[ \arccos{(x)} \right]^2 \cdot \left( -\dfrac{ 1 }{ \sqrt{ (1-x^2) } } \right) \ dx } & = & \dfrac{ \left[ \arccos{(x)} \right]^{3} } {3} \\\\ & -\int{ \left[ \arccos{(x)} \right]^2 \cdot \left( \dfrac{ 1 }{ \sqrt{ (1-x^2) } } \right) \ dx } & = & \dfrac{ \left[ \arccos{(x)} \right]^{3} } {3} \\\\ & \int{ \left[ \arccos{(x)} \right]^2 \cdot \left( \dfrac{ 1 }{ \sqrt{ (1-x^2) } } \right) \ dx } & = & -\dfrac{ \left[ \arccos{(x)} \right]^{3} } {3} \\\\ & \int{ \left[ \arccos{(x)} \right] ^2 \cdot \left( \dfrac{ 1 }{ \sqrt{ (1-x^2) } } \right) \ dx } & = & -\dfrac{1}{3} \cdot \left[ \arccos{(x)} \right]^{3} \\\\ & \int{ \left[ \arccos{(x)} \right]^2 \cdot \left( \dfrac{ 1 }{ \sqrt{ (1-x^2) } } \right) \ dx } & = & -\dfrac{1}{3} \cdot \left[\cos^{-1}{(x)}\right]^3 \\\\ \end{array} }}$$

heureka  Jun 19, 2015
#4
+26322
+10

Another approach:

$$\\ \text{Let }\theta=\cos^{-1}x\\\\ x=\cos\theta\quad\text{so }dx=-\sin\theta.d\theta\\\\ \sqrt{1-x^2}=\sqrt{1-\cos^2\theta}=\sin\theta\\\\ \text{So }\\\\ \int\frac{(\cos^{-1}x)^2}{\sqrt{1-x^2}}dx=-\int\frac{\theta^2.\sin\theta}{\sin\theta}d\theta=-\int\theta^2d\theta=-\frac{\theta^3}{3}=-\frac{(\cos^{-1}x)^3}{3}$$

There is a constant to be added of course.

.

Alan  Jun 19, 2015
#5
+78553
+10

Here's one more way using  u-du substitution.......

∫ (arccos(x))^2 / √(1 - x^2)   dx

Let  u   = arccos(x)      du = -1/ √(1 - x^2)   dx    implies that.....-du = 1/ √(1 - x^2)   dx

So we have......

-∫ u^2 du =     -(1/3)u^3  + C    =  -(1/3)[arccos(x)] + C

CPhill  Jun 19, 2015
#6
+90968
+5

Thanks Heureka and Chris,

I really like Alan's solution here.

It is so elegant.

And, once placed in front of me, it is so 'obvious'    LOL

What is the chances of it being obvious (to me) next time without Alan answering first.

I am trying to think about how something like this can be used in this question.

http://web2.0calc.com/questions/integral_4

I have included both these questions into the "Great Answers to Learn from" sticky thread

Melody  Jun 20, 2015

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