Question Bank [2022] — Maths STD 12 Science — Question
Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]2 Marks
Question
$\int x^2 \sqrt{ a ^2-x^6} d x$
✓
Answer
$ \text { Let } I =\int x^2 \sqrt{ a ^2-x^6} d x$
$=\int \sqrt{ a ^2-\left(x^3\right)^2} \cdot x^2 d x $
Put $x^3=t$
$ \therefore 3 x ^2 dx = dt$
$\therefore x ^2 dx =\frac{1}{3} dt$
$\therefore I =\frac{1}{3} \int \sqrt{ a ^2- t ^2} dt$
$=\frac{1}{3}\left[\frac{ t }{2} \sqrt{ a ^2- t ^2}+\frac{ a ^2}{2} \sin ^{-1}\left(\frac{ t }{ a }\right)\right]+ c$
$\therefore I =\frac{1}{6}\left[x^3 \sqrt{ a ^2-x^6}+ a ^2 \sin ^{-1}\left(\frac{x^3}{ a }\right)\right]+ c $
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