1. Integration without Trig Sub

How would I work the integral:
x^3 * sqrt(x^2 -9) dx

without using trig substitution? My professor said that it is a way, but I don't see it!

2. Originally Posted by latavee
How would I work the integral:
x^3 * sqrt(x^2 -9) dx

without using trig substitution? My professor said that it is a way, but I don't see it!
Split the integrand up $\displaystyle u=x^2$, and $\displaystyle dv=x\sqrt{x^2-9} \;dx$ then use integration by parts.

CB

3. THanks!

AH! Never thought of that! Thank you!

4. How about the substitution $\displaystyle u=\surd(x^2-9)$ or equivalently $\displaystyle x^2=u^2+9$? Then $\displaystyle x\,\mathrm dx=u\,\mathrm du$ and you end up with a polynomial in $\displaystyle u$ to integrate.

5. I would do $\displaystyle \int x^3\sqrt{x^2- 9}dx$ by writing it as $\displaystyle \int \left(x^2\sqrt{x^2-9}l\right)\left(xdx\right)$ and using the substitution $\displaystyle u= x^2- 9$. Then du= 2xdx so $\displaystyle \frac{1}{2}du= xdx$ and $\displaystyle x^2= u+ 9$. That gives $\displaystyle \int x^3\sqrt{x^2- 9}dx= \frac{1}{2}\int (u+ 9)\sqrt{u}du$$\displaystyle = \frac{1}{2}\int u^{3/2}+ 9u^{1/2} du$

6. I'm going to try some of these ideas out! and I will see if I can come up with the answer that I had using trig substitution/or nearly the same! Thanks guys!