Question

Find the following integral : $\int (1-x)\ \sqrt{x} \, dx$

Answer 1

$\textbf{Given:}$

$\int\,(1-x)\sqrt{x}\,dx$

$=\int\,(\sqrt{x}-x\sqrt{x})\,dx$

$=\int\,(x^{\frac{1}{2}}-x^{\frac{3}{2}})\,dx$

$\text{Using the formula, we get}$

$\boxed{\bf\int\,x^n\,dx=\frac{x^{n+1}}{n+1}+c}$

$=\frac{x^{\frac{3}{2}}}{\frac{3}{2}}-\frac{x^{\frac{5}{2}}}{\frac{5}{2}}+c$

$=\frac{2}{3}x^{\frac{3}{2}}-\frac{2}{5}x^{\frac{5}{2}}+c$

$\therefore\int\,(1-x)\sqrt{x}\,dx=\frac{2}{3}x^{\frac{3}{2}}-\frac{2}{5}x^{\frac{5}{2}}+c$

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Answer 2

$\huge{\bigstar}{ \green{ \mathfrak{ \underline{ \underline{Question}}}}}{\bigstar}$

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$Find \:the\ following\ integral\ : \\ \int (1-x)\ \sqrt{x} \, dx$

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$\huge{\bigstar}{ \green{ \mathfrak{ \underline{ \underline{Answer}}}}}{\bigstar}$

$\boxed{\red{\bold{Simplifying:}}}$

$\purple{\implies \int (1-x)\sqrt{x}\ dx}$

$\purple{\implies}\int (1-x)x^{\frac{1}{2}}dx$

$\purple{\implies}\int (x^{\frac{1}{2}} - x^{\frac{3}{2}})dx$

$\blue{\implies \int x^{\frac{1}{2}} dx - \int x^{\frac{3}{2}} dx}$

$\boxed{\red{\bold{Using\: property \: \int x^n dx = \displaystyle{\frac{x^{n+1}}{n+1}}}}}$

$\green{\implies \int x^{\frac{1}{2}} dx - \int x^{\frac{3}{2}} dx = \displaystyle{\frac{x^{\frac{1}{2} +1}}{\frac{1}{2}+1}} - \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1 }+c}$

$\green{\boxed{\implies {\boxed{\displaystyle{\frac{2x^{\frac{3}{2}}}{3}} - \frac{2x^{\frac{5}{2}}}{5} +c}}}}$

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