The integral of dx/x^2 sqrt(x^2 + 9) from sqrt(3) to cube root of 3Using trigonometric substitution



1 Answer | Add Yours

embizze's profile pic

Posted on

Evaluate `int_(sqrt(3))^(root(3)(3))(dx)/(x^2sqrt(x^2+9))` :

Let `x=3tanu` . Then `dx=3sec^2udu` .

Now `sqrt(x^2+9)=sqrt(9tan^2u+9)=sqrt(9(tan^2u+1))=3secu` ; also `u=tan^(-1)(x/3)` .

So ignoring the limits of integration for the moment we have:

`int (dx)/(x^2sqrt(x^2+9))=int(3sec^2udu)/(9tan^2u(3secu))`



This can be evaluated as:


Substituting for u and evaluating at the limits of integration:

`=(-sqrt(x^2+9))/(9x)|_sqrt(3)^(root(3)(3))`  ** `-csc(tan^(-1)(x/3))=sqrt(x^2+9)/x` **





We’ve answered 288,557 questions. We can answer yours, too.

Ask a question