Science:Math Exam Resources/Courses/MATH110/April 2016/Question 02 (d)/Solution 1

Recall that the derivative of $\cos (x)$ is $-\sin (x)$. So the derivative of $-\cos (x)$ is $\sin (x)$. Hence,$-\cos (x)+c$ is the antiderivative of $\sin (x)$. Moreover, the Power Rule states that the derivative of $x^a$ is $a{x}^{a-1}$ so in particular, the derivative of $x^{1/2}$ is $\frac{1}{2}{x}^{1/2-1}=\frac{1}{2}{x}^{-1/2}$. So as $x^{1/2}$ is the antiderivative of $\frac{1}{2}{x}^{-1/2}$, it follows that $2/\sqrt{x}=2x^{-1/2}=4\times \frac{1}{2}{x}^{-1/2}$ and that the antiderivative of $2/\sqrt{x}$ is $4x^{1/2}+c=4\sqrt{x}+c.$ Thus, the antiderivative of $\sin (x)+2/\sqrt{x}$ is $-\cos (x)+4\sqrt{x}+c.$