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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 :
$integral from negative 1 to 1 of space straight e to the power of straight x space dx$

અહી, f(x) = e^x વિધેય [-1, 1] માં સતત છે.

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

પ્રત્યેક ઉપાંતરાલની લંબાઈ $straight h space equals space fraction numerator straight b space minus space straight a over denominator straight n end fraction space equals space fraction numerator 1 space plus space 1 over denominator straight n end fraction space equals space 2 over straight n$ થશે.

આથી nh = 2

અહીં, a = -1, b = 1, f(x) = e^{x

$straight f open parentheses straight a space plus space ih close parentheses space equals space straight f open parentheses negative 1 space plus space ih close parentheses space space space space space space space space space space space space space space equals space straight e to the power of negative 1 space plus space ih end exponent space space space space space space space space space space space space space space equals space straight e to the power of negative 1 end exponent space times space straight e to the power of ih space equals space straight e to the power of ih over straight e$}

હવે, વ્યાખ્યા પ્રમાણે,

$integral from negative 1 to 1 of space straight e to the power of straight x space dx space equals space limit as straight h rightwards arrow 0 of space straight h space sum from straight i equals space 1 to straight n of space straight f space open parentheses negative 1 space plus space ih close parentheses$

$equals space limit as straight h rightwards arrow 0 of space straight h space sum from straight i space equals space 1 to straight n of space straight e to the power of ih over straight e equals space limit as straight h rightwards arrow 0 of space straight h over straight e space open square brackets straight e to the power of straight h space plus space straight e to the power of 2 straight h end exponent space plus space straight e to the power of 3 straight h end exponent space plus space...... plus straight e to the power of nh close square brackets equals space limit as straight h rightwards arrow 0 of space straight h over straight e space open square brackets fraction numerator straight e to the power of straight h space open parentheses straight e to the power of nh space minus space 1 close parentheses over denominator straight e to the power of straight h space minus space 1 end fraction close square brackets equals space limit as straight h rightwards arrow 0 of space 1 over straight e space fraction numerator straight e to the power of straight h space open parentheses straight e squared space minus space 1 close parentheses over denominator open parentheses begin display style fraction numerator straight e to the power of straight h space minus space 1 over denominator straight h end fraction end style close parentheses end fraction space space space space space space space open parentheses because space nh space equals space 2 close parentheses equals space 1 over straight e space fraction numerator straight e to the power of 0 space open parentheses straight e squared space minus space 1 close parentheses over denominator log subscript straight e space straight e end fraction space equals space fraction numerator straight e squared space minus space 1 over denominator straight e end fraction space equals space bold e bold space bold minus bold space bold e to the power of bold minus bold 1 end exponent$

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

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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નિયત સંકલન

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