Oblicz całki nieoznaczone.

[Wierzba]

1. \(\int (|x|+1) dx\)

2. \(\int min (x,x^2) dx\)

3. \(\int |1-x^2| dx\)

4x \(\int e^{|x|} dx\)

[kamil13151]

1. \(\int (|x|+1) dx = \frac{x^2 sgn(x)}{2}+x+C\)

4. \(\int e^{|x|} dx=sgn(x)e^{sgn(x)x}+C\)

[radagast]

2. \(\int min (x,x^2) dx= \begin{cases} \frac{1}{3}x^3 +C\ \ \ dla\ \ \ 0<x<1\\ \frac{1}{2}x^2 +D \ \ \ wpp \end{cases}\)

[octahedron]

1. \(\int|x|+1\,dx={\{x<0 \Rightarrow \int -x+1\,dx=-\frac{x^2}{2}+x+C\\x\ge 0 \Rightarrow \int x+1\,dx=\frac{x^2}{2}+x+C}\)

2. \(\int min (x,x^2)\,dx={\{x\in[0,1]\Rightarrow \int x^2\,dx=\frac{x^3}{3}+C\\x\in R\setminus[0,1]\Rightarrow \int x\,dx=\frac{x^2}{2}+C}\)

3. \(\int|1-x^2|\,dx={\{x\in[-1,1]\Rightarrow \int 1-x^2\,dx=-\frac{1}{3}x^3+x+C\\x\in R\setminus[-1,1]\Rightarrow \int x^2-1\,dx=\frac{1}{3}x^3-x+C}\)

4. \(\int e^{|x|}\,dx={\{x<0\Rightarrow \int e^{-x}\,dx=-e^{-x}+C\\x\ge 0\Rightarrow \int e^x\,dx=e^x+C}\)