Question

Evaluate: $\int 2^{2^{2^x}}{2}^{2^x}{2}^{x}dx$

• A. $\frac{1}{(\log 2{)}^3}{2}^{2^{2^x}}+C$
• B. $\frac{1}{(\log 2{)}^3}{2}^{2^x}+C$
• C. $\frac{1}{(\log 2{)}^2}{2}^{2^x}+C$
• D. $\frac{1}{(\log 2{)}^4}{2}^{2^{2x}}+C$

Answer 1

Answer: (A)

Let $I=\int 2^{2^{2^x}}{2}^{2^x}{2}^{x}dx$
Let $2^{2^{2^x}}=t\Rightarrow 2^{2^{2^x}}{2}^{2^x}{2}^x(\log 2{)}^{3}dx=dt$
$\Rightarrow I=\int \frac{1}{(\log 2{)}^3}dt$
$=\frac{1}{(\log 2{)}^3}t+C$
$=\frac{1}{(\log 2{)}^3}{2}^{2^{2^x}}+C$