$$\bbox[black, 5px, border: 2px solid green]  {\int u \text{ dv} = u \cdot v \:-\: \int v \cdot \text{ du}} $$

Ingat juga integral berikut:

\(\int \dfrac{1}{x} \text{ dx} = \ln |x| + c\)

\(\int e^x \text{ dx} = e^x + c\)

Contoh 01

Tentukan \(\int \ln x \cdot 3x\text{ dx}\)

Misal:

\(u = \ln x\) maka \(\frac{\text{du}}{\text{dx}} = \frac{1}{x}\)

\(\text{dv} = 3x\text{ dx}\) maka \(v = \int 3x\text{ dx}\)

\(v = \frac{3}{2}x^2\)

\(\int u \text{ dv} = u \cdot v \:-\: \int v \cdot \text{ du}\)

\(\int \ln x \cdot 3x\text{ dx} = \ln x \cdot \dfrac{3}{2}x^2 \:-\: \int \dfrac{3}{2}x^2 \cdot \dfrac{1}{x} \text{dx}\)

\(\int \ln x \cdot 3x\text{ dx} = \dfrac{3}{2}x^2 \cdot \ln x \:-\: \int \dfrac{3}{2}x\:\text{dx}\)

\(\int \ln x \cdot 3x\text{ dx} = \dfrac{3}{2}x^2 \cdot \ln x \:-\: \dfrac{3}{2}\cdot \dfrac{1}{2}x^2 + c\)

\(\int \ln x \cdot 3x\text{ dx} = \dfrac{3}{2}x^2 \cdot \ln x \:-\: \dfrac{3}{4}x^2 + c\)

Contoh 02

Tentukan \(\int 3x^2 \cdot e^x\text{ dx}\)

Misal:

\(u = 3x^2\) maka \(\frac{\text{du}}{\text{dx}} = 6x\)

\(\text{dv} = e^x\text{ dx}\) maka \(v = \int e^x\text{ dx}\)

\(v = e^x\)

\(\int u \text{ dv} = u \cdot v \:-\: \int v \cdot \text{ du}\)

\(\int 3x^2 \cdot e^x\text{ dx} = 3x^2 \cdot e^x \:-\: \int e^x \cdot 6x \text{ dx}\)

\(\int 3x^2 \cdot e^x\text{ dx} = 3x^2 \cdot e^x \:-\: \color{cyan}\int 6x \cdot e^x \text{ dx}\)

Untuk \(\color{cyan}\int 6x \cdot e^x \text{ dx}\) kita hitung sendiri dulu menggunakan integral parsial juga

Misal:

\(u = 6x\) maka \(\frac{\text{du}}{\text{dx}} = 6\)

\(\text{dv} = e^x\text{ dx}\) maka \(v = \int e^x\text{ dx}\)

\(v = e^x\)

\(\color{cyan}\int 6x \cdot e^x \text{ dx} = 6x \cdot e^x \:-\: \int e^x \cdot 6 \text{ dx}\)

\(\color{cyan}\int 6x \cdot e^x \text{ dx} = 6x \cdot e^x \:-\: 6e^x + c\)

\(\int 3x^2 \cdot e^x\text{ dx} = 3x^2 \cdot e^x \:-\: (6x \cdot e^x \:-\: 6e^x ) + c\)

\(\int 3x^2 \cdot e^x\text{ dx} = 3x^2 \cdot e^x \:-\: 6x \cdot e^x  + 6e^x + c\)