a. Evaluate the indefinite integral of (x) ln(x) dx. b. Evaluate the indefinite integral of (...

Here, we have to evaluate the given functions:

Part A.)

{eq}\displaystyle\int xlnx\ dx\\\\ {/eq}

Applying integration by parts method to do the further calculation:

{eq}=u\displaystyle\int v\ dx-\displaystyle\int \left [ \dfrac{d}{dx}(u)\displaystyle\int v\ dx \right ]dx\\\\ {/eq}

{eq}=lnx\displaystyle\int x\ dx-\displaystyle\int \left [ \dfrac{d}{dx}(lnx)\displaystyle\int x\ dx \right ]\\\\ =lnx\left ( \dfrac{x^2}{2} \right )-\displaystyle\int \left [ \dfrac{1}{x}\cdot \dfrac{x^2}{2} \right ]dx\\\\ =\dfrac{x^2}{2}lnx-\dfrac{1}{2}\displaystyle\int x\ dx\\\\ =\dfrac{x^2}{2}lnx-\dfrac{1}{2}\cdot \dfrac{x^2}{2}+c\\\\ =\dfrac{x^2}{2}lnx-\dfrac{1}{4}x^2+c {/eq}

c is the constant of integration.

Part B.)

{eq}\displaystyle\int \dfrac{x}{x^2-1}dx\\\\ {/eq}

Let:

{eq}x^2-1=z\\\\ 2x\ dx=dz\\\\ x\ dx=\dfrac{dz}{2} {/eq}

{eq}=\displaystyle\int \dfrac{1}{z}\cdot \dfrac{dz}{2}\\\\ =\dfrac{1}{2}ln\left | z \right |+c\\\\ =\dfrac{1}{2}ln\left | x^2-1 \right |+c {/eq}

c is the constant of integration.

Part C.)