Integral a^x situation: int [2^(x^3) x^2] dx

[mathtwit]

\(\displaystyle \huge\int{2^x^3 x^2} dx\)

I concluded that the answer should be

\(\displaystyle \huge\frac{1}{3} 2^x^3 + C\)

But the worked out problem says

\(\displaystyle \huge\frac{1}{3} \frac{1}{ln2} 2^x^3 + C\)

The 1/ln2 part is what confuses me, where does it come from?

 

[pka]

Here is why:
\(\displaystyle \L b > 0\quad \& \quad y = b^{f(x)} \quad \Rightarrow \quad y' = f'(x)b^{f(x)} \ln (b).\)

 

[mathtwit]

Understood. When I worked it out it seemed to me it should be ln(2) rather than 1/ln(2) but I will try it again maybe I am missing something obvious as usual.

Edit:

I had forgotten about this one rule

\(\displaystyle \L\int{a^x dx} = \frac{a^x}{ln a} + C\)

Cheers