Integrate $f(x)=\dfrac{x-1}{x^2+1}$

Question

I was working myself through old exams and stumbled across the following task:

Calculate $$\int_1^3{\dfrac{x-1}{x^2+1}dx}$$

At first I thought that I could substitute something but I really don't get this to work.

I would be very happy if someone could help me with this one.

Greetings, Finn

Answer 1

$$\displaystyle \int{\dfrac{x-1}{x^2+1}dx}=\int{\dfrac{x}{x^2+1}dx}-\int{\dfrac{1}{x^2+1}dx}=\ln(x^2+1)/2-\arctan(x)$$

Answer 2

split your integral into $$\frac{1}{2}\int \frac{2x}{x^2+1}dx-\int\frac{1}{x^2+1}dx$$

Answer 3

Consider two integrals $\int \frac{x}{x^2+1 }dx$ and $\int \frac{1}{x^2+1}dx$

Answer 4

Compute $$\left(\frac12\ln(x^2+1)-\arctan x\right)'$$