find the indefinite integral. use c to represent an arbitrary constant. ∫(√x - e^x) dx =

Explanation:

Step1: Split the integral

$\int(\sqrt{x}-e^{x})dx=\int\sqrt{x}dx-\int e^{x}dx$

Step2: Rewrite $\sqrt{x}$ and integrate

Since $\sqrt{x}=x^{\frac{1}{2}}$, and $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), then $\int x^{\frac{1}{2}}dx=\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}=\frac{2}{3}x^{\frac{3}{2}}$

Step3: Integrate $e^{x}$

We know that $\int e^{x}dx=e^{x}+C$

Step4: Combine the results

$\int\sqrt{x}dx-\int e^{x}dx=\frac{2}{3}x^{\frac{3}{2}}-e^{x}+C$

Answer:

$\frac{2}{3}x^{\frac{3}{2}}-e^{x}+C$