1. Integral Algebra

Can someone please tell me why I have a +5 and +3 in the third step?

$\int[5x^2-6x-5 + \frac{10x+3}{x^2+1}]dx$
$
5 *\frac{1}{3}x^3-6*\frac{1}{2}x^2-5x+\int\frac{10x+3}{x^2+1}dx

$

$
\frac{5}{3}x^3-3x^2-5x+5\int\frac{10x}{x^2+1}+3\int\frac{3}{x^2+1}dx$

2. Originally Posted by gammaman
Can someone please tell me why I have a +5 and +3 in the third step?

$\int[5x^2-6x-5 + \frac{10x+3}{x^2+1}]dx$
$
5 *\frac{1}{3}x^3-6*\frac{1}{2}x^2-5x+\int\frac{10x+3}{x^2+1}dx

$

$
\frac{5}{3}x^3-3x^2-5x+5\int\frac{10x}{x^2+1}+3\int\frac{3}{x^2+1}dx$
They shouldn't be there...

If you meant to write $
\frac{5}{3}x^3-3x^2-5x+5\int\frac{{\color{red}2}x}{x^2+1}+3\int\frac{{ \color{red}1}}{x^2+1}dx$
, then they should be there.

3. It's wrong.

4. Thanks for confirming my suspecision. I thought that it was wrong unless I put it there and then reduce.

5. Originally Posted by Chris L T521

If you meant to write $
\frac{5}{3}x^3-3x^2-5x+5\int\frac{{\color{red}2}x}{x^2+1}+3\int\frac{{ \color{red}1}}{x^2+1}dx$
, then they should be there.

Did I integrate wrong?

6. Originally Posted by gammaman
Did I integrate wrong?
No. All I was saying was that the 5 and 3 you had in the 3rd line of your post shouldn't be there. However, I was pointing out that $\int\frac{10x}{x^2+1}\,dx=5\int\frac{2x}{x^2+1}\,d x$ and $\int\frac{3}{x^2+1}\,dx=3\int\frac{1}{x^2+1}\,dx$ which would then imply $\int\frac{10x+3}{x^2+1}\,dx=\int\frac{10x}{x^2+1}\ ,dx+\int\frac{3}{x^2+1}\,dx=5\int\frac{2x}{x^2+1}\ ,dx+3\int\frac{1}{x^2+1}\,dx$

7. Thanks, I agree. I just wanted to make sure I wasn't going crazy