Question

Find the most general antiderivative of the function.

51. \(f\left( x \right) = {e^x} - \left( {\frac{2}{{\sqrt x }}} \right)\)

Short answer

Thus, the most general antiderivative of given function is \(G\left( x \right) = {e^x} - 4\sqrt x + C\).

Step-by-step solution

Simplify the given function

The given function is \(f\left( x \right) = {e^x} - \left( {\frac{2}{{\sqrt x }}} \right)\). Simplify the function as follows.

\(\begin{aligned}{c}f\left( x \right) &= {e^x} - \left( {\frac{2}{{\sqrt x }}} \right)\\ &= {e^x} - 2\left( {\frac{1}{{\sqrt x }}} \right)\end{aligned}\)

Obtain antiderivative given function

Let\(G\left( x \right)\)is the antiderivative of\(f\left( x \right)\).

\(\begin{aligned}{c}f\left( x \right) &= {e^x} - 2\left( {\frac{1}{{\sqrt x }}} \right)\\\frac{d}{{dx}}\left( {f\left( x \right)} \right) &= \frac{d}{{dx}}\left( {{e^x}} \right) - 2\frac{d}{{dx}}\left( {2\sqrt x } \right)\\G\left( x \right) &= {e^x} - 4\sqrt x + C\end{aligned}\)

Hence, the most general antiderivative of the given function is \(G\left( x \right) = {e^x} - 4\sqrt x + C\).