Questions: Evaluate the integral. [ int-frac23 x 3^-2 x d x ]

$Evaluate the integral. [ int-frac23 x 3^-2 x d x ]$

Transcript text: Question Evaluate the integral. \[ \int-\frac{2}{3} x 3^{-2 x} d x \]

Solution

Solution Steps

To evaluate the integral $\int -\frac{2}{3} x 3^{-2x} \, dx$, we can use integration by parts. The formula for integration by parts is $\int u \, dv = uv - \int v \, du$. We choose $u = x$ and $dv = -\frac{2}{3} \cdot 3^{-2x} \, dx$. We then find $du$ and $v$ and apply the integration by parts formula.

Step 1: Set Up the Integral

We start with the integral we want to evaluate: \[ \int -\frac{2}{3} x \cdot 3^{-2x} \, dx \]

Step 2: Apply Integration by Parts

Using the integration by parts formula $\int u \, dv = uv - \int v \, du$, we choose:

$u = x$
$dv = -\frac{2}{3} \cdot 3^{-2x} \, dx$

Calculating $du$ and $v$:

$du = dx$
To find $v$, we integrate $dv$: \[ v = \int -\frac{2}{3} \cdot 3^{-2x} \, dx = -\frac{1}{3^{2x} \log(3)} \]

Step 3: Substitute and Simplify

Now we substitute $u$, $du$, and $v$ into the integration by parts formula: \[ \int -\frac{2}{3} x \cdot 3^{-2x} \, dx = -\frac{1}{3^{2x} \log(3)} \cdot x - \int -\frac{1}{3^{2x} \log(3)} \, dx \] The remaining integral simplifies to: \[ \int -\frac{1}{3^{2x} \log(3)} \, dx = -\frac{1}{3^{2x} \log(3)} \cdot \frac{1}{\log(3)} = -\frac{1}{3^{2x} \log(3)^2} \]

Step 4: Combine Results

Combining the results, we have: \[ \int -\frac{2}{3} x \cdot 3^{-2x} \, dx = -\frac{x}{3^{2x} \log(3)} + \frac{1}{3^{2x} \log(3)^2} + C \]