I'm doing so much math I think I got a blank on the substitution method for solving integrals.
My work is scanned in the attachment. I think I'm on the right track but there is something I'm forgetting because I'm stuck in a loop.
I'm doing so much math I think I got a blank on the substitution method for solving integrals.
My work is scanned in the attachment. I think I'm on the right track but there is something I'm forgetting because I'm stuck in a loop.
$\displaystyle \int \frac{30x^2}{x^{3}-6} dx $
let $\displaystyle u = x^{3}-6$
then $\displaystyle du = 3x^2 dx \implies dx = \frac{du}{3x^2}$
so you have:
$\displaystyle \int \frac{30x^2}{x^{3}-6} dx= \int \frac{30x^2}{3x^2} \frac{1}{u}du = 10[log(u)]+C = 10[log(x^3-6)]+C $
$\displaystyle \displaystyle 5 \cdot \frac{x^5}{x^3-6} =$
$\displaystyle \displaystyle 5 \left( x^2 + \frac{6x^2}{x^3-6}\right) =$
$\displaystyle \displaystyle 5x^2 + 10 \cdot \frac{3x^2}{x^3-6}$
$\displaystyle \displaystyle 5 \int x^2 \, dx + 10\int \frac{3x^2}{x^3-6} \, dx$
$\displaystyle \displaystyle \frac{5x^3}{3} + 10\ln|x^3-6| + C$
I can follow these steps and understand them except the part where we get to the last line and we introduce ln. I know 1/u = ln|u| but where does the numerator 3x^2dx go?
Edit: Also, the integral has limits of integration 3 and 4, what do we do with this? Normally when I see limits like this I apply a formula F(b) - F(a), would this be the step after the final step shown above?
After supposing $\displaystyle u = x^3-6$, and finding $\displaystyle du = 3x^2 dx$ you get:
$\displaystyle \int \frac{30x^2}{x^{3}-6} dx= \int \frac{30x^2}{3x^2} \frac{1}{u}du$
you should be able to see that the x^2 in the numerator and denominator cancel each other other out, and 30/3 = 10...
you book should have examples on how to use the limits of integration.
$\displaystyle \displaystyle \left[\frac{5x^3}{3} + 10\ln|x^3-6|\right]^4_3$
The integral has been done so I won't bother with that. For the limits you may also leave it in terms of u and change the limits using the original subsitution:
$\displaystyle u(3) = 3^3-6 = 21$ and $\displaystyle u(4) = 4^3-6 = 58$
$\displaystyle \left[\: 10 \ln |u| \: \right]^{58}_{21}$