Question

Evaluate-
$\\ \int {2}^{ {2}^{2 ^x } } { {2}^{2 ^x } } {2}^{x}$
Please Explain in a little bit brief.​

Answer 1

Answer:

$\frac{2^{2^{2^{x}}}}{\ln^3(2)} +C$

Step-by-step explanation:

Remember:

$f(x)=a^x\\f'(x)=a^x\ln(a)$

And,

$\int a^x =\frac{a^x}{\ln(a)}$

Here,

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

Substituting,

$t=2^{2^{x}}$

$dt=\ln^2(2) \ \cdot \ 2^{2^{x}} \ \cdot \ 2^{x} \ dx$ Differentiating t w.r.t. x and using Chain Rule

You can substitute with others, but this substitution seems to yield the given equation to simplest one.

Now, the integral becomes:

$\int \frac{1}{\ln^2(2)}2^t \ dt$

$= \frac{2^{2^{2^{x}}}}{\ln^3(2)} +C$