What is the integral of 2^(2x)? tonight, exam is tomorrow

[mathnoob12345]

what is the integral of 2^(2x)? need help tonight, exam is tomorrow

teacher says the answer is: 2^(2x) / 2Ln(2)

why 2 times Ln(2)?

 

[MarkFL]

Observe that:

\(\displaystyle 2^{2x}=e^{\ln\left(2^{2x}\right)}=e^{2\ln\left(2\right)x}\)

Hence:

\(\displaystyle I=\int 2^{2x}\,dx=\int e^{2\ln\left(2\right)x}\,dx\)

Let:

\(\displaystyle u=2\ln\left(2\right)x\implies du=2\ln\left(2\right)\,dx\)

And so we have:

\(\displaystyle I=\frac{1}{2\ln\left(2\right)}\int e^u\,du\)

Can you proceed?