What is \[ \int e^{x} e^{3x} dx? \] A - HSC - SSCE Mathematics Advanced - Question 6 - 2020 - Paper 1

To evaluate the integral, we first simplify the integrand:

$$\int e^{x} e^{3x} dx = \int e^{(1 + 3)x} dx = \int e^{4x} dx.$$

Next, we apply the formula for integrating an exponential function, ( \int e^{kx} dx = \frac{1}{k} e^{kx} + C ), where ( k = 4 ):

$$\int e^{4x} dx = \frac{1}{4} e^{4x} + C.$$

Hence, the solution is: [ \frac{1}{4} e^{4x} + C ].