Penyelesaian integral dengan metode substitusi U dilakukan dengan cara memisalkan bentuk aljabar sehingga menjadi bentuk yang lebih sederhana.
Kita lihat beberapa contoh di bawah ini.
Contoh 01
\begin{equation*}
\int (2x-10)^3\:dx
\end{equation*}
Misalkan:
\begin{equation*} \begin{split} u = 2x - 10 \end{split} \end{equation*}
\begin{equation*} \begin{split} & \frac{du}{dx} = 2 \\\\ & du = 2 \: dx \\\\ & dx = \frac{du}{2} \end{split} \end{equation*}
\begin{equation*}
\begin{split}
& \int (x-2)^3\:dx \\\\
& \int u^3 \: \frac{du}{2} \\\\
& \frac{1}{2} \int u^3 \: du \\\\
& \frac{1}{2} \:.\: \frac{1}{4} \:.\: u^4 +c \\\\
& \frac{1}{8} u^4 +c \\\\
& \bbox[5px, border: 2px solid magenta] {\frac{1}{8} (2x-10)^4 +c}
\end{split}
\end{equation*}
Contoh 02
\begin{equation*}
\int (2x + 5)(x^2 + 5x)^6\:dx
\end{equation*}
Misalkan:
\begin{equation*} \begin{split} u = x^2 + 5x \end{split} \end{equation*}
\begin{equation*} \begin{split} & \frac{du}{dx} = 2x + 5 \\\\ & du = (2x + 5) \: dx \\\\ & dx = \frac{du}{(2x + 5)} \end{split} \end{equation*}
\begin{equation*}
\begin{split}
& \int (2x + 5)(x^2 + 5x)^6\:dx \\\\
& \int (2x + 5) \:.\: u^6 \: \frac{du}{(2x + 5)} \\\\
& \int \cancel {(2x + 5)} \:.\: u^6 \: \frac{du}{\cancel {(2x + 5)}} \\\\
& \int u^6 \: du \\\\
& \frac{1}{6 + 1} \:.\: u^{6 + 1} +c \\\\
& \frac{1}{7} u^7 +c \\\\
& \bbox[5px, border: 2px solid magenta] {\frac{1}{7} (x^2 + 5x)^7 +c}
\end{split}
\end{equation*}