Question

$\int (2x^2-3\sin x+5\sqrt{x})dx$ is equal to (where $C$ is the constant of integration)

• A. $\frac{2}{3}{x}^3+3\cos x+\frac{10}{3}{x}^{\frac{3}{2}}+C$
• B. $\frac{4}{3}{x}^3+3\cos x+\frac{10}{3}{x}^{\frac{3}{2}}+C$
• C. $\frac{2}{3}{x}^3-3\cos x+\frac{10}{3}{x}^{\frac{3}{2}}+C$
• D. $\frac{2}{3}{x}^3+3\cos x+\frac{10}{3}{x}^{\frac{5}{2}}+C$

Answer 1

The correct option is A $\frac{2}{3}{x}^3+3\cos x+\frac{10}{3}{x}^{\frac{3}{2}}+C$

By taking the terms separately, we get
$2\int x^{2}dx-3\int \sin xdx+5\int x^{\frac{1}{2}}dx=\frac{2x^3}{3}-3(-\cos x)+5\left(\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right)+C$
$=\frac{2}{3}{x}^3+3\cos x+\frac{10}{3}{x}^{\frac{3}{2}}+C$