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

## Question:

1. Evaluate the following indefinite integrals:

a. {eq}\displaystyle \int x\ln x \ dx {/eq}

b. {eq}\displaystyle \int \frac{x \ dx}{x^{2}-1} {/eq}

c. {eq}\displaystyle \int \frac{x \ dx}{\sqrt{x^{2}-1}} {/eq}

## Indefinite Integration:

Indefinite integral refers to integral without bounds. Thus, a constant of integration "c" is added in the end to signify an arbitrary constant.

Common formulas of integration include:

{eq}\displaystyle\int x^n\ dx=\dfrac{x^{n+1}}{n+1}+c\\\\ \displaystyle\int \dfrac{1}{x}dx=ln\left | x \right |+c\\\\ {/eq}

## Answer and Explanation:

Here, we have to evaluate the given functions:

**Part A.)**

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

Applying integration by parts method to do the...

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View this answerHere, 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.)**

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from Calculus: Tutoring Solution

Chapter 7 / Lesson 14