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Question Bank [2022] — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Question Bank [2022]4 Marks
Question
Evaluate $\int_0^{ a } x^2( a -x)^{\frac{3}{2}} d x$
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Answer

$\text { Let } I =\int_0^{ a } x^2( a -x)^{\frac{3}{2}} d x$
$\left.=\int_0^{ a }( a -x)^2[ a -( a -x)]^{\frac{3}{2}} d x \quad \ldots \ldots . . \int_0^{ a } f (x) d x=\int_0^{ a } f ( a -x) d x\right]$
$=\int_0^{ a }\left( a ^2-2 a x+x^2\right) x^{\frac{3}{2}} d x$
$=\int_0^{ a }\left( a ^2 x^{\frac{3}{2}}-2 a x^{\frac{5}{2}}+x^{\frac{7}{2}}\right) d x$
$= a ^2 \int_0^{ a } x^{\frac{3}{2}} d x-2 a \int_0^{ a } x^{\frac{5}{2}} d x+\int_0^{ a } x^{\frac{7}{2}} d x$
$= a ^2\left[\frac{x^{\frac{5}{2}}}{\frac{5}{2}}\right]_0^{ a }-2 a \left[\frac{x^{\frac{7}{2}}}{\frac{7}{2}}\right]_0^{ a }+\left[\frac{x^{\frac{9}{2}}}{\frac{9}{2}}\right]_0^{ a }$
$=\frac{2 a ^2}{5}\left[( a )^{\frac{5}{2}}-0\right]-\frac{4 a }{7}\left[( a )^{\frac{7}{2}}-0\right]+\frac{2}{9}\left[( a )^{\frac{9}{2}}-0\right]$
$=\frac{2}{5} a ^{\frac{9}{2}}-\frac{4}{7} a ^{\frac{9}{2}}+\frac{2}{9} a ^{\frac{9}{2}}$
$=\left(\frac{2}{5}-\frac{4}{7}+\frac{2}{9}\right) a ^{\frac{9}{2}}$
$=\left(\frac{126-180+70}{315}\right) a ^{\frac{9}{2}}$
$\therefore I =\frac{16}{315} a ^{\frac{9}{2}}$

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