Property 14
Definite Integral
\(\displaystyle\int f(x)\,dx=G(x)+c\)
\(a,b\in\mathbb R\)
\(\displaystyle\int\limits_{a}^{b} f(x)\,dx=\left.G(x)\right|_a^b=G(b)-G(a)\)
1.
\(\displaystyle\int\limits_{1}^{3} x\,dx=?\)
2.
\(\displaystyle\int\limits_{0}^{2} 6x^2\,dx=?\)
3.
\(\displaystyle\int\limits_{\samefrac{\pi}{6}}^{\samefrac{\pi}{3}} \cos x\,dx=?\)
4.
\(\displaystyle\int\limits_{1}^{e^4} \samefrac1x\,dx=?\)
5.
\(\displaystyle\int\limits_{1}^{3} (x^2-x)\,dx=?\)
6.
\(\displaystyle\int\limits_{0}^{\ln7} e^x\,dx=?\)
7.
\(\displaystyle\int\limits_{0}^{1} \samefrac1{1+x^2}\,dx=?\)
Property 15
\(a,b\in\mathbb R\)
\(\blacksquare\quad\displaystyle\int\limits_{a}^{b} f(x)\,dx=-\displaystyle\int\limits_{b}^{a} f(x)\,dx\)
\(\blacksquare\quad\displaystyle\int\limits_{a}^{a} f(x)\,dx=0\)
\(\blacksquare\quad\samefrac d{dx}\left(\displaystyle\int\limits_{a}^{b} f(x)\,dx\right)=0\)
\(\blacksquare\quad\displaystyle\int\limits_{a}^{b} \left(\samefrac d{dx}f(x)\right)\,dx=f(b)-f(a)\)
1.
\(\displaystyle\int\limits_{3}^{3} \left(\samefrac{2x-1}{3x+4}\right)\,dx=?\)
2.
\(\displaystyle\int\limits_{1}^{7} \samefrac{x^2}{x^3-1}\,dx+\displaystyle\int\limits_{7}^{1} \samefrac{x^2}{x^3-1}\,dx=?\)
3.
\(\samefrac d{dx}\displaystyle\int\limits_{1}^{7} \left(\samefrac{3x+1}{x^3}\right)\,dx=?\)
4.
\(\displaystyle\int\limits_{1}^{3} \left(\samefrac d{dx}\left(\samefrac{2x}{x+1}\right)\right)\,dx=?\)
5.
\(\displaystyle\int\limits_{1}^{e^3} \left(\samefrac d{dx}(\ln x)\right)\,dx=?\)