http://www.mathhelpforum.com/math-he...-4-x-2-dx.html
hi,
can anyone explain why sub X = 2 sin (theta) ?
http://www.mathhelpforum.com/math-he...-4-x-2-dx.html
hi,
can anyone explain why sub X = 2 sin (theta) ?
Inorder to do it by integration by substitution
-Have you read the complete thread?
-What's the main question you have?
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--If it is that we can substitute this directly in the formula
----than the formula is also derived like this
It's just a matter of manipulating it to use certain trigonometric identities.
If your integrand contains something of the form:
- $\displaystyle \sqrt{a^2 - x^2}$ use $\displaystyle x = a\sin \theta$ so you can use the identity: $\displaystyle 1 - \sin^2 \theta = \cos \theta$
- $\displaystyle \sqrt{x^2 - a^2}$ use $\displaystyle x = a\sec \theta$ so you can use the identity: $\displaystyle \sec^2 \theta - 1 = \tan \theta$
- $\displaystyle \sqrt{a^2 + x^2}$ use $\displaystyle x = a\tan \theta$ so you can use the identity: $\displaystyle 1 + \tan^2 \theta = \sec^2 \theta$
The question posed in this thread is an example of the first.
Or if you've dealt with hyperbolic trigonometry, you can use this identity : $\displaystyle \cosh^2(x)-\sinh^2(x)=1$
So if you have $\displaystyle \sqrt{a^2+x^2}$, you can substitute $\displaystyle x=a \sinh(t)$ and hence $\displaystyle \sqrt{a^2+x^2}=a \cosh(t)$