Factor Rule
A constant factor in front of a function remains unchanged when integrating. The factor rule allows us to pull a constant in front of the integral sign and later reintroduce it after integration.
$$ \begin{aligned}
\int\limits_{a}^{b} C \cdot f(x) \ \mathrm{d}x &= C \cdot \int\limits_{a}^{b} f(x) \ \mathrm{d}x
\end{aligned} $$
Examples:
$$ \begin{alignat}{5}
\int\limits_{a}^{b} 3 \cdot x^2 \ \mathrm{d}x &= 3 \cdot \int\limits_{a}^{b} x^2 \ \mathrm{d}x &&= 3 \cdot \dfrac{1}{3} x^3 \bigg|_{a}^{b} &&= x^3 \bigg|_{a}^{b}\\[10pt]
\int\limits_{a}^{b} 2 \cdot \sin(x) \ \mathrm{d}x &= 2 \cdot \int\limits_{a}^{b} \sin(x)\ \mathrm{d}x &&= 2 \cdot (-)\cos(x) \bigg|_{a}^{b} &&= -2 \cos(x) \bigg|_{a}^{b}
\end{alignat} $$
Sum Rule
A finite sum of functions can be integrated term by term:
$$ \begin{aligned}
\int\limits_{a}^{b} \Bigl[ f_1(x) + f_2(x) + \cdots + f_n(x)\Bigr] \ \mathrm{d}x &= \int\limits_{a}^{b} f_1(x) \ \mathrm{d}x +\int\limits_{a}^{b} f_2(x) \ \mathrm{d}x +\cdots +\int\limits_{a}^{b} f_n(x) \ \mathrm{d}x
\end{aligned} $$
Examples:
$$
\definecolor{lsgreen}{RGB}{79,175,152}
\definecolor{lsblue}{RGB}{16,160,205}
\definecolor{lsyellow}{RGB}{255,182,0}
\begin{alignat}{7}
\int\limits_{a}^{b} \Bigl[{\color{red}x^4} + {\color{lsblue}x^3} - {\color{lsyellow}x^2}\Bigr] \ \mathrm{d}x &= \int\limits_{a}^{b} {\color{red}x^4} \ \mathrm{d}x + \int\limits_{a}^{b} {\color{lsblue}x^3} \ \mathrm{d}x - \int\limits_{a}^{b} {\color{lsyellow}x^2} \ \mathrm{d}x &&= {\color{red}\dfrac{1}{5}x^5}\bigg|_{a}^{b} + {\color{lsblue}\dfrac{1}{4}x^4}\bigg|_{a}^{b} - {\color{lsyellow}\dfrac{1}{3}x^3}\bigg|_{a}^{b}\\[10pt]
\int\limits_{a}^{b} \Bigl[{\color{red}6x^2} + {\color{lsblue}3\sin(x)} \Bigr] \ \mathrm{d}x &= \int\limits_{a}^{b} {\color{red}6x^2} \ \mathrm{d}x + \int\limits_{a}^{b} {\color{lsblue}3\sin(x)} \ \mathrm{d}x &&= {\color{red}2x^3}\bigg|_{a}^{b} - {\color{lsblue}3\cos(x)}\bigg|_{a}^{b}
\end{alignat}
$$
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