I am helping my teenager son. So far I have been able to answer all his questions, and to solve all the problems he has had problems with. Except this one.
$\int{{3^x}{e^x}dx}$
Please, help me keep my success track and my son's confidence.
$\endgroup$2 Answers
$\begingroup$Hint: try rewriting $3^x$ as $e^{x \ln 3}$.
$\endgroup$ 8 $\begingroup$Why not simply $\int 3^x e^x \mathrm dx = \int (3e)^x \mathrm dx = \frac{(3e)^x}{\ln(3e)} + Constant$
To see that $\int (3e)^x \mathrm dx = \frac{(3e)^x}{\ln(3e)}+C$, You can differentiate $(3e)^x$ i.e. Let
$$y = (3e)^x$$
$$\ln y = x \ln(3e)$$
$$ \mathrm dy/\mathrm dx = y \ln(3e) = (3e)^x \ln(3e)$$
$\endgroup$ 3
