Tuesday, November 4, 2014

Advanced Math Solutions – Integral Calculator, trigonometric substitution

In the previous posts we covered substitution, but standard substitution is not always enough. Integrals involving radicals for instance, we want to get rid of the square root.
Here’s how:

  • For \sqrt{a^2-bx^2}, let x=\frac{\sqrt{a}}{\sqrt{b}}\sin \left(u\right) and use the identity 1-\sin^2(u)=\cos^2(u)
  • For \sqrt{a^2+bx^2}, let x=\frac{\sqrt{a}}{\sqrt{b}}\tan \left(u\right) and use the identity 1+\tan^2(u)=\sec^2(u)
  • For \sqrt{bx^2-a^2}, let x=\frac{\sqrt{a}}{\sqrt{b}}\sec \left(u\right) and use the identity \sec^2(u)-1=\tan^2(u)

Here’s an example of tangent substitution (click here):


From here simply cancel, integrate and substitute back


Here’s an example of sine substitution (click here):


 
Here’s an example of secant substitution (click here):



We covered almost all there is to know about substitution.  In the next post we will cover inverse trigonometric functions.


Cheers,
Michal

30 comments:

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  3. can u find integral of this:
    sin(a+3x)-3sin(a+2x)+3sin(a+x)-sina
    -----------------------------------
    8sin^3(x/2)
    i cant find it on integral calculator

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  4. Thanks, my friend. Yes you covered almost about trigonometric substitution. But if you proved standard substitution then it will be a great blog.

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