Determine $\int [{(\frac{1}{2})}^{x} + x^{1 / 2}] d x$
A. $\frac{1}{\ln (2) \cdot 2^{x}} + \frac{2}{3} x^{3 / 2} + C$
B. $\frac{1}{\ln (1 / 2)} {(\frac{1}{2})}^{x} + \frac{2}{3} x^{3 / 2} + C$
$c - \frac{1}{\ln (x) \cdot 2^{x}} + \frac{2}{3} x^{3 / 2} + C$
D. $\frac{1}{x + 1} {(\frac{1}{2})}^{x + 1} + \frac{2}{3} x^{3 / 2} + C$
E. $- \frac{1}{\ln (2) \cdot 2^{x}} + \frac{2}{3} x^{5 / 2} + C$

Answer

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Answer

Final Answer: $\frac{1}{\ln (2) \cdot 2^{x}} + \frac{2}{3} x^{3 / 2} + C$

Steps

Step 1 ：The integral is a sum of two terms, so we can integrate each term separately.

Step 2 ：The first term is a simple exponential function, and the second term is a power function.

Step 3 ：The integral of an exponential function is the same function divided by the natural logarithm of its base.

Step 4 ：The integral of a power function is the function with its exponent increased by 1, divided by the new exponent.

Step 5 ：Therefore, the integral of the function is $\frac{1}{\ln (2) \cdot 2^{x}} + \frac{2}{3} x^{3 / 2} + C$ .

Step 6 ：Final Answer: $\frac{1}{\ln (2) \cdot 2^{x}} + \frac{2}{3} x^{3 / 2} + C$