Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Integrals

Long Answer Type

21.

Evaluate the following integrals as the limit of a sum
$integral subscript 0 superscript 3 left parenthesis straight x squared plus 2 straight x right parenthesis space dx$

111 Views

22.

Evaluate the following integrals as the limit of a sum
$integral subscript 0 superscript 3 left parenthesis 2 straight x squared plus 3 right parenthesis space dx$

102 Views

23.

Evaluate $integral subscript 2 superscript 3 space straight x cubed space dx$ as the limit of a sum.

153 Views

24. Evaluate
$integral subscript straight a superscript straight b space straight e to the power of straight x space dx$ as limit of sum.

Comparing $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space with space integral subscript straight a superscript straight b space straight e to the power of straight x dx comma space space we space get comma$
   $straight f left parenthesis straight x right parenthesis space equals space straight e to the power of straight x$
$therefore$ $straight f left parenthesis straight a right parenthesis space equals space straight e to the power of straight a comma space space straight f left parenthesis straight a plus straight h right parenthesis space equals space straight e to the power of straight a plus straight h end exponent comma$
   $straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight e to the power of straight a plus 2 straight h end exponent space space...... space space straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight e to the power of straight a plus left parenthesis straight n minus 1 right parenthesis straight h end exponent$
Now,    $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
$rightwards double arrow$ $integral subscript straight a superscript straight b straight e to the power of straight x dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight e to the power of straight a plus straight e to the power of straight a plus straight h end exponent plus straight e to the power of straight a plus 2 straight h end exponent plus... plus straight e to the power of straight a plus left parenthesis straight n minus 1 right parenthesis straight h end exponent right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below space straight h space straight e to the power of straight a space open square brackets fraction numerator 1 left parenthesis straight e to the power of nh minus 1 end exponent right parenthesis over denominator straight e to the power of straight h minus 1 end exponent end fraction close square brackets$ $open square brackets because space space space straight S subscript straight n space equals space straight a fraction numerator left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction close square brackets$

   $equals space Lt with straight h rightwards arrow 0 below space straight h space straight e to the power of straight a space open square brackets fraction numerator straight e to the power of straight b minus straight a end exponent minus 1 over denominator straight e to the power of straight h minus 1 end fraction close square brackets space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space space nh space equals space straight b minus straight a right square bracket$

   $equals space straight e to the power of straight a space left parenthesis straight e to the power of straight b minus straight a end exponent minus 1 right parenthesis space Lt with straight h rightwards arrow 0 below space open parentheses fraction numerator 1 over denominator begin display style fraction numerator straight e to the power of straight h minus 1 over denominator straight h end fraction end style end fraction close parentheses$

   $equals space left parenthesis straight e to the power of straight b minus straight e to the power of straight a right parenthesis. space 1 over 1 space equals space straight e to the power of straight b space minus space straight e to the power of straight a$

97 Views

25.

Evaluate $integral subscript negative 1 end subscript superscript 1 space straight e to the power of straight x space dx$ as the limit of a sum.

231 Views

26.

Evaluate $integral subscript 0 superscript 2 straight e to the power of straight x space dx$ as the limit of a sum.

107 Views

27.

Evaluate $integral subscript 0 superscript 1 straight e to the power of 2 minus 3 straight x end exponent space dx$ as a limit of a sum.

101 Views

Short Answer Type

28.

Evaluate $integral subscript 0 superscript 3 straight e to the power of straight x space dx$ as the limit of sums.

90 Views

29.

Evaluate $integral subscript 0 superscript 4 left parenthesis straight x plus straight e to the power of 2 straight x end exponent right parenthesis space dx$ as the limit of a sum.