Question

Integrate:
$\int 2^{{\log }_{4}x}dx$

Answer 1

$\int 2^{{\log }_{4}x}dx$

As we know that,

${\log }_{a^b}c=\frac{1}{b}{\log }_{a}c$

$\therefore \int 2^{{\log }_{4}x}dx=\int 2^{{\log }_{2^2}x}dx$

$\Rightarrow =\int 2^{\frac{1}{2}{\log }_{2}x}dx$

$\Rightarrow =\int 2^{{\log }_2{x}^{\frac{1}{2}}}dx$

$\Rightarrow =\int x^{\frac{1}{2}}dx(\because a^{{\log }_{a}b}=b)$

$\Rightarrow =\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C=\frac{2}{3}{x}^{\frac{3}{2}}+C$

Hence $\int 2^{{\log }_{4}x}dx=\frac{2}{3}{x}^{\frac{3}{2}}+C$