Question

(int x^{9}ln x dx)

Explanation:

Step1: Choose Integration by Parts

Recall integration by parts formula: $\int u \, dv = uv - \int v \, du$. Let $u = \ln x$ (so that $du = \frac{1}{x} dx$) and $dv = x^9 dx$ (so we can find $v$ by integrating $dv$).

Step2: Find \( v \)

Integrate $dv = x^9 dx$: $v = \int x^9 dx = \frac{x^{10}}{10}$ (using power rule $\int x^n dx = \frac{x^{n + 1}}{n + 1} + C$ for $n
eq -1$).

Step3: Apply Integration by Parts

Substitute into the formula: $\int x^9 \ln x dx = uv - \int v \, du = \ln x \cdot \frac{x^{10}}{10} - \int \frac{x^{10}}{10} \cdot \frac{1}{x} dx$. Simplify the integral: $\int \frac{x^{10}}{10x} dx = \frac{1}{10} \int x^9 dx$.

Step4: Integrate the Remaining Term

Integrate $\frac{1}{10} \int x^9 dx$: $\frac{1}{10} \cdot \frac{x^{10}}{10} + C = \frac{x^{10}}{100} + C$.

Step5: Combine Terms

Put it all together: $\frac{x^{10} \ln x}{10} - \frac{x^{10}}{100} + C$. We can factor out $\frac{x^{10}}{100}$ to get $\frac{x^{10}(10 \ln x - 1)}{100} + C$ or leave it as $\frac{x^{10} \ln x}{10} - \frac{x^{10}}{100} + C$.

Answer:

$\frac{x^{10} \ln x}{10} - \frac{x^{10}}{100} + C$ (or equivalent factored form)