Property 23
Integral of the Defined Function
\(\displaystyle\int\limits_{a}^{b} f(x)\,dx=\displaystyle\int\limits_{a}^{c} f(x)\,dx+\displaystyle\int\limits_{c}^{b} f(x)\,dx\)
\(\displaystyle\int\limits_{a}^{b} f(x)\,dx=\displaystyle\int\limits_{a}^{c} f(x)\,dx+\displaystyle\int\limits_{c}^{b} f(x)\,dx\)
Integral of the defined functions according to the critical points
\(\blacksquare\quad f(x)=\begin{cases}h(x)&x\ge a\\g(x)&x<a\end{cases}\)
a critical point
\(\blacksquare\quad\begin{gathered}f(x)=|g(x)|\\g(a)=0\end{gathered}\)
a critical point
\(\blacksquare\quad\begin{gathered}f(x)=\operatorname{sgn}(g(x))\\g(a)=0\end{gathered}\)
a critical point
\(\blacksquare\quad\begin{gathered}f(x)=\left[\!\left[ g(x)\right]\!\right]\\g(a)\in\mathbb Z\end{gathered}\)
a critical point
1.
\(f(x)=\begin{cases}x^2&x>2\\2x&x\le2\end{cases}\)
\(\Rightarrow\displaystyle\int\limits_{0}^{3} f(x)\,dx=?\)
2.
\(f(x)=\begin{cases}x+1&x\ge1\\x^3&x<1\end{cases}\)
\(\Rightarrow\displaystyle\int\limits_{0}^{2} f(x)\,dx=?\)
3.
\(f(x)=\begin{cases}4x^2&x<-2\\3&x=-2\\-2x&x>-2\end{cases}\)
\(\Rightarrow\displaystyle\int\limits_{-3}^{1} f(x)\,dx=?\)
4.
\(\displaystyle\int\limits_{-2}^{4} |x|\,dx=?\)