Question

What is the integral of (7x+2)/(x^2+4) dx?

Answer 1

My problem is I can't find anything to substitute of manipulate to make 7x + 4 go away.

Answer 2

Note that \[\begin{aligned}\int\frac{7x+2}{x^2+4}\,dx &= \int \frac{7x}{x^2+4} + \frac{2}{x^2+4}\,dx\\ &= 7\int\frac{x}{x^2+4}\,dx + 2\int\frac{1}{x^2+4}\,dx\end{aligned}\]Do you know how to proceed from here? :-)

Answer 3

That helps a lot! I tried splitting it, but not like that. Thank you, I believe I can take it from here.

Answer 4

... I thought I did it right, but no luck
. Online still says im wrong. I'll post my answer.

Answer 5

\[7\ln(x^2+4)+2\tan^{-1} (x/4)+C\]

Answer 6

Opps hold on I didn't take care of the constant changes

Answer 7

You have the right idea, but it isn't fully correct. You're missing a factor of \(\dfrac{1}{2}\) in the first term (it pops up when you do substitution). Also, for the second integral note that\[\large \begin{aligned} 2\int\frac{1}{x^2+4}\,dx &= 2\int\frac{1}{4\left(\frac{x^2}{4}+1\right)}\,dx\\ &= \frac{1}{2}\int\frac{1}{\frac{x^2}{4}+1}\,dx\\ &= \frac{1}{2}\int\frac{1}{\left(\frac{x}{2}\right)^2+1}\,dx \end{aligned}\]Can you fix your mistakes from here? :-)

Answer 8

Got it! Just made a problem with what constants were out in front.
Final answer of \[\frac{ 7 }{ 2 } \ln(x^2+4)+\tan^{-1} (x/2)+C\]

Answer 9

There you go. :-)

Answer 10

Thanks for the help!