Question

Evaluate the integral\(\int {\frac{{{{\rm{2}}^{\sqrt {\rm{x}} }}}}{{\sqrt {\rm{x}} }}} {\rm{dx}}\).

Short answer

The integral value of the given equation is\(\int {\frac{{{{\rm{2}}^{\sqrt {\rm{x}} }}}}{{\sqrt {\rm{x}} }}} {\rm{dx = }}\frac{{{\rm{2 \times }}{{\rm{2}}^{\sqrt {\rm{x}} }}}}{{{\rm{ln2}}}}{\rm{ + C}}\).

Step-by-step solution

Expand the equation.

Given: \(\int {\frac{{{{\rm{2}}^{\sqrt {\rm{x}} }}}}{{\sqrt {\rm{x}} }}} {\rm{dx}}\)

Substitute

\(\begin{array}{c}{\rm{2}}\int {{{\rm{2}}^{\rm{u}}}} {\rm{du}}\\{\rm{u = }}\sqrt {\rm{x}} {\rm{\;}}\\{\rm{du = }}\frac{{\rm{1}}}{{{\rm{2}}\sqrt {\rm{x}} }}{\rm{dx}}\\{\rm{2du = }}\frac{{\rm{1}}}{{\sqrt {\rm{x}} }}{\rm{dx}}\end{array}\)

Evaluate the equation.

\(\begin{array}{c}{\rm{ = }}\frac{{{\rm{2 \times }}{{\rm{2}}^{\rm{u}}}}}{{{\rm{ln2}}}}{\rm{ + C}}\\{\rm{ = }}\frac{{{\rm{2 \times }}{{\rm{2}}^{\sqrt {\rm{x}} }}}}{{{\rm{ln2}}}}{\rm{ + C}}\end{array}\)

Therefore, the integral value of the given equation is\(\int {\frac{{{{\rm{2}}^{\sqrt {\rm{x}} }}}}{{\sqrt {\rm{x}} }}} {\rm{dx = }}\frac{{{\rm{2 \times }}{{\rm{2}}^{\sqrt {\rm{x}} }}}}{{{\rm{ln2}}}}{\rm{ + C}}\).