Question

Integral 2^( x-1) dx

Answer 1

$\int\limits {2^{x-1}} \, dx \\ u=x-1 \\ du = dx \\ \int\limits{2^{u}} \, du \\ \\ 2^u = e^{u \cdot ln(2)} \\ \\ \int\limits{e^{u \cdot ln(2)}} \, du \\ \\ v = u \cdot ln(2) \\ dv = ln(2)du \Rightarrow du = \frac{1}{ln(2)}dv \\ \\ \frac{1}{ln(2)}\int\limits{e^{v}} \, dv = \frac{1}{ln(2)}\cdot e^v = \frac{1}{ln(2)}\cdot e^{u \cdot ln(2)} =\frac{1}{ln(2)}\cdot e^{(x-1)\cdot ln(2)}=\frac{2^{x-1}}{ln(2)} \\ \\ \int\limits {2^{x-1}} \, dx = \frac{2^{x-1}}{ln(2)} + C$