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Evaluate \(\rm \int x^2\sqrt x dx\)1. \(\rm \frac{7}{2} x^{\frac{7}{2}} +c\)2. \(\rm \frac{2}{7} x^{\frac{7}{2}} +c\)3. \(\rm \frac{2}{5} x^{\frac{7}{2}} +c\)4. \(\rm \frac{2}{7} x^{\frac{5}{2}} +c\) |
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Answer» Correct Answer - Option 2 : \(\rm \frac{2}{7} x^{\frac{7}{2}} +c\) Concept: \(\rm \int x^n dx = \frac{x^{n+1}}{n+1}+c\) Calculation: Let I = \(\rm \int x^2\sqrt x dx\) = \(\rm \int x^2 x^{\frac{1}{2}} dx \) = \(\rm \int x^{2+\frac{1}{2}} dx \) (∵ xaxb = xa+b) = \(\rm \int x^{\frac{5}{2}} dx \) = \(\rm \frac{ x^{\frac{5}{2} + 1}}{\frac{5}{2} + 1} +c\) = \(\rm \frac{2}{7} x^{\frac{7}{2}} +c\) |
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