Question

Evaluate: ${\int }_2^3\frac{xdx}{x^2+1}$

Answer 1

The given integral can be written in the form $\int \frac{f^{\prime }\left(x\right)}{f\left(x\right)}dx$ by multiplying 2 in the numerator and dinominator. The formula of integral is $\int \frac{f^{\prime }\left(x\right)}{f\left(x\right)}dx$=$ln\left(f\right(x\left)\right)+c$.

By using this formula,${\int }_2^3\frac{xdx}{x^2+1}={\frac{ln[x^2+1]}{2}}_2^3+c$

=$\frac{1}{2}\left(ln\right[10]-ln[5\left]\right)=\frac{1}{2}ln\left[2\right]$