What you think about the this test of CalcMentor of this platform?
I tested "integrate x^2 * ln(x)" into CalcMentor here on this site and find below what I got in return. There is a PDF result attached below, which was exported/downloaded directly from this site for free.
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### 📊 CalcMentor AI — Mathematical Solution
**Problem:** $$\int x^2 \cdot \ln(x) \, dx$$
**Answer:** $$\frac{x^{3} \cdot \left(3 \ln{\left(x \right)} - 1\right)}{9} + C$$
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#### 🔍 STEP-BY-STEP SOLUTION
**Step 1: Choose $u$ and $dv$ (Integration by Parts)** To solve this using integration by parts, we apply the LIATE rule. Let's choose:
$$u = \ln(x)$$
$$dv = x^{2} \, dx$$
**Step 2: Differentiate $u$ and Integrate $dv$** * Differentiating $u = \ln(x)$ gives:
$$\frac{du}{dx} = \frac{1}{x} \implies du = \frac{1}{x} \, dx$$
* Integrating $dv = x^{2} \, dx$ gives:
$$v = \frac{x^{3}}{3}$$
**Step 3: Apply the Integration by Parts Formula** The integration by parts formula is:
$$\int u \, dv = u \cdot v - \int v \, du$$
Plugging in our assigned values, we get:
$$\int x^{2} \cdot \ln(x) \, dx = \ln(x) \cdot \frac{x^{3}}{3} - \int \frac{x^{3}}{3} \cdot \frac{1}{x} \, dx$$
**Step 4: Simplify the Integral** Simplifying the terms inside the remaining integral on the right-hand side:
$$\int x^{2} \cdot \ln(x) \, dx = \frac{x^{3} \ln(x)}{3} - \int \frac{x^{2}}{3} \, dx$$
**Step 5: Integrate the Remaining Term** Now, evaluate the straightforward integral of $\frac{x^2}{3}$:
$$\int x^{2} \cdot \ln(x) \, dx = \frac{x^{3} \ln(x)}{3} - \frac{x^{3}}{9} + C$$
**Step 6: Factor Out the Common Term** To find a common denominator and clean up the expression, we can factor out $\frac{x^3}{9}$:
$$\int x^{2} \cdot \ln(x) \, dx = \frac{x^{3}}{9} \left(3 \ln(x) - 1\right) + C$$
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**Final Verified Answer:** $$\frac{x^{3} \cdot \left(3 \ln{\left(x \right)} - 1\right)}{9} + C$$
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* **Generated:** June 4, 2026 at 02:06 AM
* **Source:** [derivativecalculus.com/calcmentor.html](https://derivativecalculus.com/calcmentor.html)
📎 CalcMentor AI — Solution.pdf — Click to download
***
### 📊 CalcMentor AI — Mathematical Solution
**Problem:** $$\int x^2 \cdot \ln(x) \, dx$$
**Answer:** $$\frac{x^{3} \cdot \left(3 \ln{\left(x \right)} - 1\right)}{9} + C$$
---
#### 🔍 STEP-BY-STEP SOLUTION
**Step 1: Choose $u$ and $dv$ (Integration by Parts)** To solve this using integration by parts, we apply the LIATE rule. Let's choose:
$$u = \ln(x)$$
$$dv = x^{2} \, dx$$
**Step 2: Differentiate $u$ and Integrate $dv$** * Differentiating $u = \ln(x)$ gives:
$$\frac{du}{dx} = \frac{1}{x} \implies du = \frac{1}{x} \, dx$$
* Integrating $dv = x^{2} \, dx$ gives:
$$v = \frac{x^{3}}{3}$$
**Step 3: Apply the Integration by Parts Formula** The integration by parts formula is:
$$\int u \, dv = u \cdot v - \int v \, du$$
Plugging in our assigned values, we get:
$$\int x^{2} \cdot \ln(x) \, dx = \ln(x) \cdot \frac{x^{3}}{3} - \int \frac{x^{3}}{3} \cdot \frac{1}{x} \, dx$$
**Step 4: Simplify the Integral** Simplifying the terms inside the remaining integral on the right-hand side:
$$\int x^{2} \cdot \ln(x) \, dx = \frac{x^{3} \ln(x)}{3} - \int \frac{x^{2}}{3} \, dx$$
**Step 5: Integrate the Remaining Term** Now, evaluate the straightforward integral of $\frac{x^2}{3}$:
$$\int x^{2} \cdot \ln(x) \, dx = \frac{x^{3} \ln(x)}{3} - \frac{x^{3}}{9} + C$$
**Step 6: Factor Out the Common Term** To find a common denominator and clean up the expression, we can factor out $\frac{x^3}{9}$:
$$\int x^{2} \cdot \ln(x) \, dx = \frac{x^{3}}{9} \left(3 \ln(x) - 1\right) + C$$
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**Final Verified Answer:** $$\frac{x^{3} \cdot \left(3 \ln{\left(x \right)} - 1\right)}{9} + C$$
***
* **Generated:** June 4, 2026 at 02:06 AM
* **Source:** [derivativecalculus.com/calcmentor.html](https://derivativecalculus.com/calcmentor.html)