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Question
\[\int\limits_0^4 x\sqrt{4 - x} dx\]
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Solution
\[Let, I = \int_0^4 x\sqrt{4 - x} d x\]
\[ = \int_0^4 \left( 4 - x \right)\sqrt{4 - 4 + x} d x\]
\[ = \int_0^4 \left( 4 - x \right)\sqrt{x} d x\]
\[ = \int_0^4 4\sqrt{x} - x^\frac{3}{2} dx\]
\[ = \left[ 8\frac{x^\frac{3}{2}}{3} \right]_0^4 - \left[ \frac{2 x^\frac{5}{2}}{5} \right]_0^4 \]
\[ = \frac{64}{3} - \frac{64}{5}\]
\[ = \frac{128}{15}\]
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