Question

निश्चित समाकलन में गुणधर्म का उपयोग करते हुए प्रश्न में समाकलन का मान ज्ञात कीजिए-

${\displaystyle {\int }_0^4 \left|x-1\right| dx}$

Answer 1

${\displaystyle {\int }_0^4 \left|x-1\right| dx}$

परिभाषित करना,

${\displaystyle \left|x-1\right|=\{\begin{array}{lll}-(x-1)& ifx-1<0& \text{or}x<1\\ x-1& ifx-1\ge 0& \text{or}x\ge 1\end{array}}$

${\displaystyle {\int }_0^1\left|x-1\right| dx+{\int }_1^4 \left|x-1\right| dx}$

${\displaystyle {\int }_0^1-(x-1) dx+{\int }_1^4 (x-1) dx}$

${\displaystyle =-{[\frac{x^2}{2}-x]}_0^1+{[\frac{x^2}{2}-x]}_1^4}$

${\displaystyle =[(\frac{1}{2}-1)-0]+(\frac{16}{2}-4)-(\frac{1}{2}-1)}$

${\displaystyle =\frac{1}{2}+4+\frac{1}{2}}$

${\displaystyle =\frac{1+8+1}{2}=5}$