Tentukan hasil dari \( \int x^4 \ln \left( \frac{25}{x} \right) \ dx = \cdots \ ? \).
Pembahasan:
Pertama kita bisa sederhanakan fungsi dalam integralnya terlebih dahulu. Dari sifat logaritma natural, \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \), sehingga:
\begin{aligned} \int x^4 \ln \left( \frac{25}{x} \right) \ dx = \int x^4 \left( \ln 25 - \ln x \right) \ dx \end{aligned}
Dengan demikian, kita peroleh berikut:
\begin{aligned} \int x^4 \ln \left( \frac{25}{x} \right) \ dx &= \int x^4 \left( \ln 25 - \ln x \right) \ dx \\[8pt] &= \int x^4 \ln 25 \ dx - \int x^4 \ln x \ dx \\[8pt] &= \ln 25 \int x^4 \ dx - \int x^4 \ln x \ dx \\[8pt] &= \ln 25 \cdot \frac{1}{5}x^5 - \frac{1}{5}x^5 \left(\ln x - \frac{1}{5} \right) + C \\[8pt] &= \frac{x^5}{5} \left( \ln 25 - \ln x + \frac{1}{5}\right) + C \end{aligned}
Ingat bahwa:
\begin{aligned} \int x^n \ln x \ dx = \frac{1}{n+1}x^{n+1} \left( \ln x - \frac{1}{n+1} \right) + C \end{aligned}
Sehingga:
\begin{aligned} \int x^n \ln x \ dx &= \frac{1}{n+1}x^{n+1} \left( \ln x - \frac{1}{n+1} \right) + C \\[8pt] \int x^4 \ln x \ dx &= \frac{1}{4+1}x^{4+1} \left( \ln x - \frac{1}{4+1} \right) + C \\[8pt] &= \frac{1}{5} x^5 \left(\ln x - \frac{1}{5} \right) + C \end{aligned}