this question is designed to be answered without a calculator.\n$int(10^{x}-\frac{1}{sqrt{x}})dx=$\n$10^{x}-s…

Explanation:

Step1: Split the integral

Use the property $\int(f(x)-g(x))dx=\int f(x)dx-\int g(x)dx$. So, $\int(10^{x}-\frac{1}{\sqrt{x}})dx=\int 10^{x}dx-\int x^{-\frac{1}{2}}dx$.

Step2: Integrate $10^{x}$

The integral of $a^{x}$ is $\frac{a^{x}}{\ln a}+C$. For $a = 10$, $\int 10^{x}dx=\frac{10^{x}}{\ln 10}+C_1$.

Step3: Integrate $x^{-\frac{1}{2}}$

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). Here $n=-\frac{1}{2}$, so $\int x^{-\frac{1}{2}}dx=\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+C_2=\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+C_2 = 2\sqrt{x}+C_2$.

Step4: Combine the results

$\int(10^{x}-\frac{1}{\sqrt{x}})dx=\frac{10^{x}}{\ln 10}-2\sqrt{x}+C$ (where $C = C_1 - C_2$).

Answer:

$\frac{10^{x}}{\ln 10}-2\sqrt{x}+C$