Question

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) –\(\frac{4x^{5/2}}{5}+\frac{2x^{7/2}}{7}+C\)

This question was addressed to me in homework.

This intriguing question comes from Integration as an Inverse Process of Differentiation in section Integrals of Mathematics – Class 12

Answer 1

Correct answer is (c) \(\frac{4x^{5/2}}{6}+\frac{2x^{7/2}}{7}+C\)

The best explanation: To find \(\int (2+x)x\sqrt{x} dx\)

\(\int \,(2+x)x\sqrt{x} \,dx=\int \,2x\sqrt{x}+x^{5/2} \,dx\)

\(\int \,(2+x)x\sqrt{x} \,dx=\int \,2x^{3/2} dx + \int x^{5/2} dx\)

\(\int \,(2+x)x\sqrt{x} \,dx=\frac{2x^{3/2+1}}{3/2+1}+\frac{x^{5/2+1}}{5/2+1}\)

\(\int \,(2+x)x\sqrt{x} \,dx=\frac{4x^{5/2}}{5}+\frac{2x^{7/2}}{7}+C\)