Thread: Integration using substitution

The following definite integral can be evaluated to give the answer A - B where A and B are found from substituting the limits of integration.



What are A and B?

Having a difficult time showing how to integrate this to get the answer
3ln(7x^2 + 5x + 3). The denominator can be represented by u and du would be 14x + 5. With this, I divide and multiply it all by 14 I believe to get the 3 thats in the answer. This will give me the integral of 3/u. According to integration rules, 1/u is equal to the ln of u. Thus i get 3ln(u) and sub back in the original values. Then you evaluate at 3 and 0 to get A and B. Is this correct? If it doesnt make sense then let me know I can try to clarify. Thankss!

I think this is what they are getting at... we have

$\displaystyle \int_{0}^{3}\frac{42x^2 + 15}{7x^2 + 5x + 3}dx$

And decide to let $\displaystyle u = 7x^2 + 5x + 3$
Then $\displaystyle du = (14x + 5)dx$ so our integral becomes...

$\displaystyle 3*\int_{?}^{?}\frac{du}{u}$

Notice that I have left the limits of integration out. They are no longer 0 and 3. One way to prevent errors would just be to always explicitly write
$\displaystyle \int_{x = 0}^{x = 3}\frac{42x^2 + 15}{7x^2 + 5x + 3}dx$

But that's tedious.

So we will see that, when x = 0, u = 3 (from our choice of u sub above).
Likewise, when x = 3, u = ...

THOSE are the "A" and "B" to which they refer.

Thanks for the help. I think I got it.

You are welcome.