Question

Evaluate integral: integral (3^(-2xln3))

Answer 1

\[\int3^{-2x\ln3}~dx\]
Let \(u=-2x\ln3\), so that \(-\dfrac{1}{2\ln3}du=dx\), or \(-\dfrac{1}{2\ln3}du=dx\).
\[-\frac{1}{2\ln3}\int3^u~du\]
Now use some exponent/log properties:
\[3^u=e^{\ln3^u}=e^{u\ln3}\]
\[-\frac{1}{2\ln3}\int e^{u\ln3}~du\]
Let \(t=u\ln3\), so that \(\dfrac{1}{\ln3}dt=du\):
\[-\frac{1}{2\ln3}\cdot\frac{1}{\ln3}\int e^t~dt\]
\[-\frac{1}{2(\ln3)^2}\int e^t~dt\]