Bab 14 Pengamiran

$\begin{array}{l}\text{Katakan }u=3x+5\\ \frac{du}{dx}=3\\ dx=\frac{du}{3}\\ \\ {{\displaystyle \int \left(3x+5\right)}}^{3}dx\\ ={\displaystyle \int u^3}\frac{du}{3}\leftarrow \boxed{\begin{array}{l}\text{gantikan }3x+5=u\\ \text{dan }dx=\frac{du}{3}\end{array}}\\ =\frac{1}{3}{\displaystyle \int u^3}du\\ =\frac{1}{3}\left(\frac{u^4}{4}\right)+c\\ =\frac{1}{3}\left(\frac{{\left(3x+5\right)}^4}{4}\right)+c\leftarrow \boxed{\begin{array}{l}\text{ ganti balik}\\ u=3x+5\end{array}}\\ =\frac{{\left(3x+5\right)}^4}{12}+c\end{array}$

$\begin{array}{l}{\displaystyle \int {\left(ax+b\right)}^n}=\frac{{\left(ax+b\right)}^{n+1}}{\left(n+1\right)a}+c\\ \\ \text{Oleh itu, }\\ {\displaystyle \int {\left(3x+5\right)}^{3}dx}=\frac{{\left(3x+5\right)}^4}{4\left(3\right)}+c\\ =\frac{{\left(3x+5\right)}^4}{12}+c\end{array}$

$\begin{array}{l}\text{(a) }{\displaystyle \int \frac{2}{7}}{\left(5-x\right)}^{-4}dx=\frac{2{\left(5-x\right)}^{-3}}{7\left(-3\right)\left(-1\right)}+c\\ =\frac{2}{21{\left(5-x\right)}^3}+c\end{array}$

$\begin{array}{l}\text{(b) }{\displaystyle \int \frac{2}{3{\left(9x-2\right)}^5}dx}={\displaystyle \int \frac{2{\left(9x-2\right)}^{-5}}{3}}dx\\ =\frac{2{\left(9x-2\right)}^{-4}}{3\left(-4\right)\left(9\right)}+c\\ =-\frac{2}{108{\left(9x-2\right)}^4}+c\\ =-\frac{1}{54{\left(9x-2\right)}^4}+c\end{array}$