Question

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Answer 1

$A)\frac{4^{3x}}{6\ln(2)}+C$

1) Let's evaluate this integral using the due properties. We can start out by raising 4 to the 3rd power before actually integrating:

$\begin{gathered} \int\:4^{3x}dx\Rightarrow\int\:(4^3)^xdx \\ \int \:64^xdx \end{gathered}$

2) Now, let's apply the following property, and rewrite the logarithm we have just found:

$\begin{gathered} \int a^xdx=\frac{a^x}{\ln a} \\ \int64^xdx=\frac{64^x}{\ln(64)} \\ \frac{64^{x}}{\operatorname{\ln}(64)}=\frac{2^{6x}}{\ln(2)^6}\Rightarrow\frac{2^{6x}}{6\ln(2)} \end{gathered}$

3) Let's rewrite those power and that log:

$\begin{gathered} \frac{2^{6x}}{2\cdot\:3\ln\left(2\right)}=\frac{2^{6x-1}}{3\ln\left(2\right)} \\ \\ \int\:4^{3x}dx=\frac{2^{6x-1}}{3\operatorname{\ln}(2)}+C \\ \end{gathered}$

Note that among the options there is one equivalent option, then the answer is:

$A)\int64^xdx=\frac{4^{3x}}{6\operatorname{\ln}(2)}+C$