An integration involving inverse trig.

[Charismaztex]

Homework Statement

The question is integrate

$$\int_0^1 (2+\frac{1}{x+1}+\frac{2x+1}{x^2+4}) dx$$

Homework Equations

$$\int_0^1 (2+\frac{1}{x+1}+\frac{2x+1}{x^2+4}) dx =2 + ln(\frac{5}{2}) +\frac{1}{2}arctan(\frac{1}{2})$$ (yes, it's a prove question)

The Attempt at a Solution

The first two parts I have no problem but I am not quite sure how to integrate the

$$\frac{2x+1}{x^2+4}$$ part. I know it has to do with arctan but I'm not quite sure about the $$2x+1$$ part.

Thanks in advance,
Charismaztex

 

[Dunkle]

Split up the last fraction so that you have $$\frac{2x}{x^{2}+4} + \frac{1}{x^{2}+4}$$. Use a simple substitution for the first fraction and then the second fraction will involve the inverse tangent. In general,

$$\int \frac{dx}{x^2+a^2} = \frac{1}{a} arctan(\frac{x}{a}) + C$$

 

[Charismaztex]

ahh! I completely missed that step! Thanks for your help :)

Charismaztex