Solving $\int e^{kx}(f(x)+f'(x))\, dx.$

Solution 1:

$$\int_1^2 e^{2x}\Big(\ln x+\frac1x\Big)\, \mathrm dx = \int_1^2 e^{2x}\ln x\,\mathrm dx +\int_1^2 \frac{e^{2x}}x\,\mathrm dx $$

Integrating by parts with $\ln x =u$,

$$\begin{align}\int_1^2 e^{2x}\Big(\ln x+\frac1x\Big)\, \mathrm dx &= \left. \frac12e^{2x}\ln x\right|_1^2 -\frac12\int_1^2 \frac{e^{2x}}x\,\mathrm dx +\int_1^2 \frac{e^{2x}}x\,\mathrm dx \\ &= \frac12e^4\ln2+\frac12\int_1^2\frac{e^{2x}}x\,\mathrm dx \\ &= \frac12e^4\ln2+\frac12\int_2^4\frac{e^u}u\,\mathrm du \\ &= \frac12e^4\ln2 +\frac12[\text{Ei}(4)-\text{Ei}(2)]\end{align}$$

As you can see, no escape from the exponential integral. But, the result can be expressed in the form of the hypergeometric and the Meijer G function.