Question

Evaluate the integral.

\(\int_1^2 {\left( {\frac{x}{2} - \frac{2}{x}} \right)} dx\)

Short answer

The value of \(\int_1^2 {\left( {\frac{x}{2} - \frac{2}{x}} \right)} dx\)is\(\frac{3}{4} - 2\ln 2\).

Step-by-step solution

Definition of the integral

Integral calculus is an area of mathematics that deals with the calculation, properties, and applications of integrals.

Evaluating the integral.

The give data is expressed as,

\(\begin{aligned}{c}\int_1^2 {\left( {\frac{x}{2} - \frac{2}{x}} \right)} dx &= \int_1^2 {\frac{x}{2}} dx - \int_1^2 {\frac{2}{x}} dx\\ &= \frac{1}{2}\int_1^2 x dx - 2\int_1^2 {\frac{1}{x}} dx\\ &= \frac{1}{2}\left( {\frac{{{x^2}}}{2}} \right)_1^2 - 2(\ln |x|)_1^2\\ &= \frac{1}{2}\left( {\frac{{{2^2}}}{2} - \frac{{{1^2}}}{2}} \right) - 2(\ln 2 - \ln 1)\\ &= \frac{1}{2}\left( {\frac{{4 - 1}}{2}} \right) - 2(\ln 2 - 0)\\ &= \frac{3}{4} - 2\ln 2\end{aligned}\)

Hence the value of \(\int_1^2 {\left( {\frac{x}{2} - \frac{2}{x}} \right)} dx\) is \(\frac{3}{4} - 2\ln 2\).