Kindly help me with this evaluate the integral \gamma \binom{2}{1} \frac{2x}{{x}^(2) + 1} dx ​

Question

Kindly help me with this

evaluate the integral

`\gamma \binom{2}{1} \frac{2x}{{x}^(2) + 1} dx`
​

Answer 1

Answer: ln5 - ln2 ≈ 0.919

Explanation:

`\int\limits^2_1 {(2x)/(x^2+1)} \, dx\qquad \qquad =\int\limits^2_1 {(x^2+1)^(-1)} \, 2xdx`

Let u = x² + 1 → du = 2x

`\text{Then we have:}\quad \int\limits^2_1 {u^(-1)} \, du \qquad =ln|u|\bigg|^2_1`

Substitute u = x² + 1

`ln|x^2+1|\bigg|^2_1\quad =ln|2^2+1|-ln|1^2+1|\quad = ln|5|-ln|2|`