Question

$\int (2^x+2^{-x}{)}^{2}dx=$

• A. $\frac{4^x}{\log 4}-\frac{4^{-x}}{\log 4}+2x+c$
• B. $\frac{4^x}{\log 4}+\frac{4^{-x}}{\log 4}+2x+c$
• C. $4x+4^{-x}+c$
• D. $4x-4+c$

Answer 1

Answer: (A)

$\frac{4^x}{\log 4}-\frac{4^{-x}}{\log 4}+2x+c$

$\int (2^x+2^{-x}{)}^{2}dx$
$=\int (2^{2x}+2^{-2x}+2)dx$
$=\int (4^x+4^{-x}+2)dx$
$=\int (4^x+{\left(\frac{1}{4}\right)}^x+2)dx$
$=\frac{4^x}{\log 4}+\frac{{\frac{1}{4}}^x}{\log \left(\frac{1}{4}\right)}+2x+c$
$=\frac{4^x}{\log 4}-\frac{4^{-x}}{\log 4}+2x+c$