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

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