Questions: Solve ∫ x^11 ln(x) dx using Integration by Parts. Use u=ln(x) and v'=x^11.

Transcript text: Solve $\int x^{11} \ln (x) d x$ using Integration by Parts. Use $u=\ln (x)$ and $v^{\prime}=x^{11}$.

Solution

Solution Steps

To solve the integral $\int x^{11} \ln (x) \, dx$ using Integration by Parts, we will use the formula $\int u \, dv = uv - \int v \, du$. Here, we are given $u = \ln(x)$ and $dv = x^{11} \, dx$. We need to find $du$ and $v$ and then apply the formula.

Step 1: Identify $ u $ and $ dv $

Given the integral $\int x^{11} \ln(x) \, dx$, we use Integration by Parts with: \[ u = \ln(x) \] \[ dv = x^{11} \, dx \]

Step 2: Compute $ du $ and $ v $

Differentiate $ u $ and integrate $ dv $: \[ du = \frac{d}{dx} \ln(x) = \frac{1}{x} \, dx \] \[ v = \int x^{11} \, dx = \frac{x^{12}}{12} \]

Step 3: Apply the Integration by Parts Formula

Using the formula $\int u \, dv = uv - \int v \, du$: \[ \int x^{11} \ln(x) \, dx = \ln(x) \cdot \frac{x^{12}}{12} - \int \frac{x^{12}}{12} \cdot \frac{1}{x} \, dx \]

Step 4: Simplify the Integral

Simplify the expression: \[ \int x^{11} \ln(x) \, dx = \frac{x^{12} \ln(x)}{12} - \frac{1}{12} \int x^{11} \, dx \] \[ \int x^{11} \ln(x) \, dx = \frac{x^{12} \ln(x)}{12} - \frac{1}{12} \cdot \frac{x^{12}}{12} \] \[ \int x^{11} \ln(x) \, dx = \frac{x^{12} \ln(x)}{12} - \frac{x^{12}}{144} \]

Step 5: Combine and Simplify the Result

Combine the terms: \[ \int x^{11} \ln(x) \, dx = \frac{x^{12} \ln(x)}{12} - \frac{x^{12}}{144} \] \[ \int x^{11} \ln(x) \, dx = \frac{x^{12} (12 \ln(x) - 1)}{144} \]