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નિયત સંકલન

નીચે આપેલાં નિયત સંકલિતને સરવાળાના લક્ષ સ્વરૂપે મેળવો :

$integral from 1 to 3 of space open parentheses straight x squared space plus space straight x close parentheses space dx$

નીચે આપેલાં નિયત સંકલિતને સરવાળાના લક્ષ સ્વરૂપે મેળવો :
$integral from negative 1 to 1 of space straight e to the power of straight x space dx$

નીચે આપેલાં નિયત સંકલિતને સરવાળાના લક્ષ સ્વરૂપે મેળવો :
$integral from 0 to 1 of space straight e to the power of 2 minus 3 straight x end exponent space dx$

અહી, વિધેય f(x) = e^{2 - 3x} એ [0, 1] માં સતત છે.

[0, 1] નું સમાન લંબાઈના n ઉપાંતરાલોમાં વિભાજન કરતાં,

પ્રત્યેક ઉપાંતરાલની લંબાઈ $straight h space equals space fraction numerator straight b space minus space straight a over denominator straight n end fraction space equals space fraction numerator 1 space minus space 0 over denominator straight n end fraction space equals space 1 over straight n$

આથી nh = 1

અહીં, a = 0, b = 1 અને

$straight f open parentheses straight a space plus space ih close parentheses space equals space straight f open parentheses 0 space plus space ih close parentheses space equals space straight f open parentheses ih close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight e to the power of 2 minus 3 ih end exponent space equals space straight e squared space times space straight e to the power of negative 3 ih end exponent space$

જેમ $straight n space rightwards arrow space infinity$ તેમ $straight h space rightwards arrow space 0.$

હવે, $integral from 0 to 1 of space straight e to the power of 2 space minus space 3 straight x end exponent space dx$

$equals space limit as straight h rightwards arrow 0 of space straight h space sum from straight i space equals space 1 to straight n of space straight f open parentheses straight a space plus space ih close parentheses equals space space limit as straight h rightwards arrow 0 of space straight h space sum from straight i space equals space 1 to straight n of space straight e squared space times space straight e to the power of negative 3 ih end exponent equals space space limit as straight h rightwards arrow 0 of space straight h space times space straight e squared space open square brackets straight e to the power of negative 3 straight h end exponent space plus space open parentheses straight e to the power of negative 3 straight h end exponent close parentheses squared space plus space..... plus space open parentheses straight e to the power of negative 3 straight h end exponent close parentheses to the power of straight n close square brackets equals space space limit as straight h rightwards arrow 0 of space straight h space times space straight e squared space open square brackets fraction numerator straight e to the power of negative 3 straight h end exponent space open parentheses open parentheses straight e to the power of negative 3 straight h end exponent close parentheses to the power of straight n space minus space 1 close parentheses over denominator straight e to the power of negative 3 straight h end exponent space minus space 1 end fraction close square brackets equals space space limit as straight h rightwards arrow 0 of space fraction numerator negative 3 straight h over denominator straight e to the power of negative 3 straight h end exponent space minus space 1 end fraction space cross times space fraction numerator straight e squared over denominator negative 3 end fraction space cross times space open parentheses straight e to the power of negative 3 open parentheses nh close parentheses end exponent space minus space 1 close parentheses space cross times space straight e to the power of negative 3 straight h end exponent equals space space limit as straight h rightwards arrow 0 of space straight h space fraction numerator negative 3 straight h over denominator straight e to the power of negative 3 straight h end exponent space minus 1 end fraction space cross times space fraction numerator straight e squared over denominator negative 3 end fraction space cross times space open parentheses straight e to the power of negative 3 space cross times space 1 end exponent space minus space 1 close parentheses space cross times space straight e to the power of negative 3 straight h end exponent space space space space space space open parentheses because space nh space equals space 1 close parentheses$

$equals space fraction numerator 1 over denominator log subscript straight e space straight e end fraction space cross times space fraction numerator straight e squared over denominator negative 3 end fraction space open parentheses straight e to the power of negative 3 end exponent space minus space 1 close parentheses space cross times space straight e to the power of 0 equals space 1 over 1 space cross times space fraction numerator 1 over denominator negative 3 end fraction space open parentheses straight e to the power of negative 1 end exponent space minus space straight e squared close parentheses space cross times space 1 bold equals bold space bold 1 over bold 3 bold space open parentheses bold e to the power of bold 2 bold space bold minus bold space bold e to the power of bold minus bold 1 end exponent close parentheses$

નીચે આપેલાં નિયત સંકલિતને સરવાળાના લક્ષ સ્વરૂપે મેળવો :
$integral from 1 to 2 of space 3 to the power of straight x space dx$

નીચે આપેલાં નિયત સંકલિતને સરવાળાના લક્ષ સ્વરૂપે મેળવો :
$integral from log subscript straight e space 2 to log subscript straight e space 5 of space straight e to the power of straight x space dx$

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