Find \(\int (2+x)x\sqrt{x} dx\).

Category: QuestionsFind \(\int (2+x)x\sqrt{x} dx\).
Editor">Editor Staff asked 11 months ago

Find \(\int (2+x)x\sqrt{x} dx\).
(a) \(\frac{4x^{5/2}}{5}+\frac{2x^{7/2}}{9}+C\)
(b) \(\frac{4x^{5/2}}{5}-\frac{2x^{7/2}}{7}+C\)
(c) \(\frac{4x^{5/2}}{6}+\frac{2x^{7/2}}{7}+C\)
(d) –\(\frac{4x^{5/2}}{5}+\frac{2x^{7/2}}{7}+C\)
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This intriguing question comes from Integration as an Inverse Process of Differentiation in section Integrals of Mathematics – Class 12
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1 Answers
Editor">Editor Staff answered 11 months ago

Correct answer is (c) \(\frac{4x^{5/2}}{6}+\frac{2x^{7/2}}{7}+C\)
The best explanation: To find \(\int (2+x)x\sqrt{x} dx\)
\(\int \,(2+x)x\sqrt{x} \,dx=\int \,2x\sqrt{x}+x^{5/2} \,dx\)
\(\int \,(2+x)x\sqrt{x} \,dx=\int \,2x^{3/2} dx + \int x^{5/2} dx\)
\(\int \,(2+x)x\sqrt{x} \,dx=\frac{2x^{3/2+1}}{3/2+1}+\frac{x^{5/2+1}}{5/2+1}\)
\(\int \,(2+x)x\sqrt{x} \,dx=\frac{4x^{5/2}}{5}+\frac{2x^{7/2}}{7}+C\)