Question

homework9: problem 5
(1 point)
find an antiderivative ( f(x) ) of ( f(x)=3 x-sqrt{x} ).
( f(x)= )

Explanation:

Step1: Recall antiderivative rules

The antiderivative of \(x^n\) (for \(n
eq - 1\)) is \(\frac{x^{n + 1}}{n+1}+C\). We will find the antiderivative of each term in \(f(x)=3x-\sqrt{x}\) separately. First, rewrite \(\sqrt{x}\) as \(x^{\frac{1}{2}}\).

Step2: Antiderivative of \(3x\)

For the term \(3x\), where \(n = 1\) in the power - rule formula. The antiderivative of \(x\) is \(\frac{x^{1 + 1}}{1+1}=\frac{x^{2}}{2}\), so the antiderivative of \(3x\) is \(3\times\frac{x^{2}}{2}=\frac{3x^{2}}{2}\).

Step3: Antiderivative of \(-\sqrt{x}\)

For the term \(-\sqrt{x}=-x^{\frac{1}{2}}\), using the power - rule with \(n=\frac{1}{2}\). The antiderivative is \(-\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}=-\frac{x^{\frac{3}{2}}}{\frac{3}{2}}=-\frac{2}{3}x^{\frac{3}{2}}\).

Step4: Combine the antiderivatives

We can ignore the constant of integration \(C\) (since we just need an antiderivative, not the most general one). So \(F(x)=\frac{3x^{2}}{2}-\frac{2}{3}x^{\frac{3}{2}}\).

Answer:

\(\frac{3}{2}x^{2}-\frac{2}{3}x^{\frac{3}{2}}\)