Calcule a derivada a seguir: \[ \int x^{4} \ln x d x \] Qual das alternativas aponta a resposta correta? a. $x^{\wedge} 5 \ln x-x^{\wedge} 5$ b. $(1 / 5) \ln x^{\wedge} 5-x^{\wedge} 5+C$ c. $x^{\wedge} 5 \ln x-x^{\wedge} 5+c$ d. $(1 / 5) x^{\wedge} 5 \ln x-(1 / 25) x^{\wedge} 5+C$ e. $x^{\wedge} 5 / 5-(1 / 25) x^{\wedge} 5+C$

Step 1 ：Choose \(u = \ln x\) and \(dv = x^4 dx\)

Step 2 ：Calculate \(du = \frac{d(\ln x)}{dx} dx = \frac{1}{x} dx\) and \(v = \int x^4 dx = \frac{x^5}{5}\)

Step 3 ：Use integration by parts formula: \(\int x^4 \ln x dx = uv - \int v du\)

Step 4 ：Plug in values: \(\int x^4 \ln x dx = \frac{x^5 \ln x}{5} - \int \frac{x^5}{5} \cdot \frac{1}{x} dx\)

Step 5 ：Calculate the remaining integral: \(\int \frac{x^5}{5} \cdot \frac{1}{x} dx = \int \frac{x^4}{5} dx = \frac{x^5}{25}\)

Step 6 ：Combine terms: \(\int x^4 \ln x dx = \frac{x^5 \ln x}{5} - \frac{x^5}{25} + C\)

Step 7 ：\(\boxed{(1 / 5) x^{5} \ln x-(1 / 25) x^{5}+C}\)