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Solve: \int x- \frac {2}{x} + \sqrt {x^2} \ dx

Question:

Solve:

{eq}\int x- \frac {2}{x} + \sqrt {x^2} \ dx {/eq}

Integration's Basic Rules:

To find the integration of a monomial, use the general power rule of integrations.

  • {eq}\int x^{n}\ dx=\frac{x^{n+1}}{n+1}+C {/eq}

Where,

  • {eq}n {/eq} is any constant and {eq}C {/eq} is the constant of integration.

The integration of {eq}1 {/eq} over {eq}x {/eq} is a natural logarithm function i.e. {eq}\ln (x) {/eq}.

Answer and Explanation:

Given data:

{eq}\int x- \frac {2}{x} + \sqrt {x^2} \ dx =? {/eq}

The integration of each term of the above expression is:

{eq}\begin{align*} \int x- \frac {2}{x} + \sqrt {x^2} \ dx&=\int x\ dx - \int \frac {2}{x}\ dx + \int \sqrt {x^2} \ dx\\ &=\frac{x^{2}}{2} - 2\int \frac {1}{x}\ dx + \int (x^2)^{\frac{1}{2}} \ dx\\ &=\frac{x^{2}}{2} - 2\ln(x) + \int x \ dx\\ &=\frac{x^{2}}{2} - 2\ln(x) + \frac{x^{2}}{2}+C\\ &=\frac{2x^{2}}{2} - 2\ln(x)+C\\ &=x^{2}- 2\ln(x)+C\\ \end{align*} {/eq}


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