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નિયત સંકલન

નીચે આપેલાં નિયત સંકલિતને સરવાળાના લક્ષ સ્વરૂપે મેળવો :

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

અહી, વિધેય f(x) = x² + x એ [1, 3] માં સતત છે.

[1, 3] નું સમાન લંબાઈના n ઉપાંતરાલોમાં વિભાજન કરતાં,

પ્રત્યેક ઉપાંતરાલની લંબાઈ $straight h space equals space fraction numerator straight b space minus space straight a over denominator straight n end fraction space equals space fraction numerator 3 space minus space 1 over denominator straight n end fraction space equals space 2 over straight n.$

અહીં, a = 1, b = 3 અને

$straight f open parentheses straight a space plus space ih close parentheses space equals space straight f space open parentheses 1 space plus space ih close parentheses$

$equals space open parentheses 1 space plus space ih close parentheses squared space plus space open parentheses 1 space plus space ih close parentheses equals space 1 space plus space 2 ih space plus space straight i squared straight h squared space plus space 1 space plus space ih equals space straight i squared space straight h squared space plus space 3 ih space plus space 2$

હવે, વ્યાખ્યા પ્રમાણે,

$integral from 1 to 3 of space open parentheses straight x squared space plus space straight x close parentheses space dx equals space limit as straight n rightwards arrow infinity of space straight h space sum from straight i space equals space 1 to straight n of space straight f space open parentheses straight a space plus space ih close parentheses equals space limit as straight n rightwards arrow infinity of space 2 over straight n space sum from straight i space equals space 1 to straight n of space open square brackets straight i squared space straight h squared space plus space 3 ih space plus space 2 close square brackets equals space limit as straight n rightwards arrow infinity of space 2 over straight n space open square brackets straight h squared space sum from straight i space equals space 1 to straight n of space straight i squared space plus space 3 straight h space sum from straight i space equals space 1 to straight n of space straight i space plus space 2 space sum from straight i space equals space 1 to straight n of space 1 close square brackets equals space limit as straight n rightwards arrow infinity of space 2 over straight n space open square brackets 4 over straight n squared space times space straight n over 6 space open parentheses straight n space plus space 1 close parentheses space open parentheses 2 straight n space plus space 1 close parentheses space plus space 3 space times space 2 over straight n space times space straight n over 2 space open parentheses straight n space plus space 1 close parentheses space plus space 2 space times space straight n close square brackets equals space limit as straight n rightwards arrow infinity of space open square brackets 4 over 3 space open parentheses 1 space plus space 1 over straight n close parentheses space open parentheses 2 space plus space 1 over straight n close parentheses space plus space 6 space open parentheses 1 space plus space 1 over straight n close parentheses space plus space 4 close square brackets$

$equals space 4 over 3 space open parentheses 1 space plus space 0 close parentheses space open parentheses 2 space plus space 0 close parentheses space plus space 6 space open parentheses 1 space plus space 0 close parentheses space plus space 4 equals space 8 over 3 space plus space 6 space plus space 4 bold equals bold space bold 38 over bold 3$

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

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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