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 $ax^{a-1}$ 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 $\color {blue}-\cos(x)+4{\sqrt {x}}+c.$