Integration by trigonometric change of variable

[MMM]

Homework Statement

I'm trying to solve $\int\sqrt{a^2 - x^2}$
by using the substitution $x = asin\theta$

Homework Equations

##x = asin\theta

The Attempt at a Solution

$y = \int\sqrt{a^2 - a^2cos^2\theta}$
$y = a\int\cos\theta$
$y = a^2\int\cos(\theta)^2$
$y = (a^2)/2 * \int1+cos2\theta$
$y = a^2/2 + a^2/4 * sin2\theta\ + C$
$\theta = arcsin(x/a)$
$a^2/2 * arcsin(x/a) + a^2/2 * sin\theta\cos\theta\ + C$ Any help would greatly be appreciated.

The answer in the book is $(a^2/2) * arcsin(x/a) + (x/2) * \sqrt{a^2 - x^2} + C$

EDIT: I figured it out.

 

[verty]

Homework Statement

I'm trying to solve $\int\sqrt{a^2 - x^2}$
by using the substitution $x = asin\theta$

Homework Equations

##x = asin\theta

The Attempt at a Solution

$y = \int\sqrt{a^2 - a^2cos^2\theta}$
$y = a\int\cos\theta$
$y = a^2\int\cos(\theta)^2$
$y = (a^2)/2 * \int1+cos2\theta$
$y = a^2/2 + a^2/4 * sin2\theta\ + C$
$\theta = arcsin(x/a)$
$a^2/2 * arcsin(x/a) + a^2/2 * sin\theta\cos\theta\ + C$ Any help would greatly be appreciated.

The answer in the book is $(a^2/2) * arcsin(x/a) + (x/2) * \sqrt{a^2 - x^2} + C$

EDIT: I figured it out.

You have some crazy algebra happening there. You are not ready for tests or exams yet because your working out is incorrect. It must be neat and each statement must follow from the previous one. For example, this is meaningless: $y = a \int cos\theta$, that is not how an integral is written.

 

[HallsofIvy]

Homework Statement

I'm trying to solve $\int\sqrt{a^2 - x^2}$

You have started off by copying the problem wrong. It should be $\int\sqrt{a^2- x^2}dx$
Do you see the difference?

by using the substitution $x = asin\theta$

Homework Equations

##x = asin\theta

The Attempt at a Solution

$y = \int\sqrt{a^2 - a^2cos^2\theta}$

Same mistake as before.

$y = a\int\cos\theta$

I have not idea where you got this, [tex]\sqrt{a^2- a^2 cos(\theta)} is NOT equal to "$a cos(\theta)$"<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> $y = a^2\int\cos(\theta)^2$ </div> </div> </blockquote> And this is definitely not equal to the previous line!<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> $y = (a^2)/2 * \int1+cos2\theta$<br /> $y = a^2/2 + a^2/4 * sin2\theta\ + C$<br /> $\theta = arcsin(x/a)$<br /> $a^2/2 * arcsin(x/a) + a^2/2 * sin\theta\cos\theta\ + C$ Any help would greatly be appreciated.<br /> <br /> The answer in the book is $(a^2/2) * arcsin(x/a) + (x/2) * \sqrt{a^2 - x^2} + C$<br /> <br /> EDIT: I figured it out. </div> </div> </blockquote>[/tex]