Property 24
\(f:\mathbb R\longrightarrow\mathbb R\)
\(f(-x)=f(x)\quad\text{(f dual function)}\)
Dual functions are symmetrical in relation to y axis.
\(\displaystyle\int\limits_{-a}^{a} f(x)\,dx=2\displaystyle\int\limits_{0}^{a} f(x)\,dx\quad a\in\mathbb R\)
1.
\(\Rightarrow S_1+S_2=?\)
2.
\(\displaystyle\int\limits_{-4}^{4} |x|\,dx=?\)
3.
\(\displaystyle\int\limits_{-1}^{1} (x^4+x^2)\,dx=?\)
4.
\(\displaystyle\int\limits_{-\samefrac{\pi}{2}}^{\samefrac{\pi}{2}} \cos x\,dx=?\)
5.
\(\displaystyle\int\limits_{-3}^{3} (x^2+4)\,dx=?\)
6.
\(\displaystyle\int\limits_{-2}^{2} (x^2-1)\,dx=?\)
Property 25
\(f:\mathbb R\longrightarrow\mathbb R\)
\(f(-x)=-f(x)\quad\text{(f single function)}\)
Single functions are symmetrical in relation to origin.
\(\displaystyle\int\limits_{-a}^{a} f(x)\,dx=0\quad a\in\mathbb R\)
1.
\(\displaystyle\int\limits_{-9}^{9} (4x^5-7x^3+10x)\,dx=?\)
2.
\(\displaystyle\int\limits_{-\samefrac{\pi}{6}}^{\samefrac{\pi}{6}} x\cdot\cos x\,dx=?\)
3.
\(\displaystyle\int\limits_{-\pi}^{\pi} \sin^3x\,dx=?\)
4.
\(\displaystyle\int\limits_{-\samefrac{\pi}{3}}^{\samefrac{\pi}{3}} x^2\cdot\tan x\,dx=?\)
5.
\(\displaystyle\int\limits_{-2}^{2} x^2\cdot(x+1)\,dx=?\)