Question

Evaluate the integral $\displaystyle{\int_{0}^{1}\frac{x}{x+1}dx}$

1. $1-\ln{ 2}$
2. $\ln{2}$
3. 0
4. $-\ln{2}$

Answer 1

$$\displaystyle\int_{0}^{1}\dfrac{x}{x+1}dx=\displaystyle\int_{0}^{1}\dfrac{x+1-1}{x+1}dx$$

$$=\displaystyle\int_{0}^{1}dx-\displaystyle\int_{0}^{1}\dfrac{1}{x+1}dx$$

$$=[x]\begin{matrix} 1 \\ 0 \end{matrix}-\ln(|x+1|)\begin{matrix} 1 \\ 0 \end{matrix}$$

$$=1-0-(\ln(|1+1|)-\ln(|0+1|))$$

$$=1-\ln2$$

1. $1-\ln2$