Evaluating integrals

[Kcashew]

I am stuck on trying to take the integral of (-0.25x+1)/(x^2 + 4).

I believe that I should use arctan, but I do not know what to do about the numerator.

Where should I go from here?

 

[MarkFL]

If presented with an integral of the form:

[MATH]I=\int\frac{ax+b}{x^2+c^2}\,dx[/MATH]
I would recommend breaking it up into two integrals as follows:

[MATH]I=\frac{a}{2}\int\frac{2x}{x^2+c^2}\,dx+b\int\frac{1}{x^2+c^2}\,dx[/MATH]
Can you proceed?

 

[Kcashew]

I believe that I can.

The answer I have gotten is this:

0.25ln(x)-0.125ln(x^2+4)+0.5tan^-1(x/2)+C

 

[MarkFL]

Let me check:

[MATH]I=\int\frac{-0.25x+1}{x^2+4}\,dx=-\frac{1}{8}\int\frac{2x}{x^2+2^2}\,dx+\int\frac{1}{x^2+2^2}\,dx=-\frac{1}{8}\ln(x^2+4)+\frac{1}{2}\arctan\left(\frac{x}{2}\right)+C[/MATH]
Where did you get the first term in your anti-derivative?