New Notes Blog: `int (x^3 - 2sqrt(x))/x dx` Find the general indefinite integral.

Friday, 2 May 2014

Given $\displaystyle\int{\left(\frac{{{{x}}^{{3}}-{2}\sqrt{{{x}}}}}{{x}}\right)}{\left.{d}{x}\right.}$

Simplify the expression by dividing each term by x.

$\displaystyle=\int{\left({{x}}^{{2}}-{2}{{x}}^{{-\frac{{1}}{{2}}}}\right)}{\left.{d}{x}\right.}$

Integrate each term.

$\displaystyle={\left(\frac{{{x}}^{{3}}}{{3}}\right)}-\frac{{{\left({2}{x}\right)}}^{{\frac{{1}}{{2}}}}}{{\frac{{1}}{{2}}}}+{C}$

Simplify.

$\displaystyle={\left(\frac{{{x}}^{{3}}}{{3}}\right)}-{4}{{x}}^{{\frac{{1}}{{2}}}}+{C}$

$\displaystyle={\left(\frac{{{x}}^{{3}}}{{3}}\right)}-{4}\sqrt{{{x}}}+{C}$

Given $\displaystyle\int{\left(\frac{{{{x}}^{{3}}-{2}\sqrt{{{x}}}}}{{x}}\right)}{\left.{d}{x}\right.}$

Simplify the expression by dividing each term by x.

$\displaystyle=\int{\left({{x}}^{{2}}-{2}{{x}}^{{-\frac{{1}}{{2}}}}\right)}{\left.{d}{x}\right.}$

Integrate each term.

$\displaystyle={\left(\frac{{{x}}^{{3}}}{{3}}\right)}-\frac{{{\left({2}{x}\right)}}^{{\frac{{1}}{{2}}}}}{{\frac{{1}}{{2}}}}+{C}$

Simplify.

$\displaystyle={\left(\frac{{{x}}^{{3}}}{{3}}\right)}-{4}{{x}}^{{\frac{{1}}{{2}}}}+{C}$

$\displaystyle={\left(\frac{{{x}}^{{3}}}{{3}}\right)}-{4}\sqrt{{{x}}}+{C}$

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