Questions: Use integration by parts to evaluate the integral. [ int 2 x ln (5 x) d x ]

Solution Steps

To solve the integral \(\int 2x \ln(5x) \, dx\) using integration by parts, we need to identify parts of the integrand that we can set as \(u\) and \(dv\). We typically choose \(u\) to be a function that simplifies when differentiated and \(dv\) to be a function that is easy to integrate. Here, we can set \(u = \ln(5x)\) and \(dv = 2x \, dx\). Then, we differentiate \(u\) to find \(du\) and integrate \(dv\) to find \(v\). Finally, we apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).

To use integration by parts, we choose: \[ u = \ln(5x) \] \[ dv = 2x \, dx \]

Differentiate \(u\) to find \(du\): \[ du = \frac{d}{dx} \ln(5x) = \frac{1}{x} \, dx \]

Integrate \(dv\) to find \(v\): \[ v = \int 2x \, dx = x^2 \]

The integration by parts formula is: \[ \int u \, dv = uv - \int v \, du \]

Substitute \(u\), \(v\), and \(du\) into the formula: \[ \int 2x \ln(5x) \, dx = x^2 \ln(5x) - \int x^2 \left(\frac{1}{x}\right) \, dx \]

Simplify the remaining integral: \[ \int x^2 \left(\frac{1}{x}\right) \, dx = \int x \, dx = \frac{x^2}{2} \]

Combine the results to get the final expression: \[ \int 2x \ln(5x) \, dx = x^2 \ln(5x) - \frac{x^2}{2} \]

Final Answer