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

Short Answer Type

191.

Show that:
$integral subscript straight pi over 4 end subscript superscript straight pi over 4 end superscript open vertical bar sin space straight x close vertical bar dx space equals space 2 minus square root of 2$

120 Views

Long Answer Type

192.

Show that:
$integral subscript negative 1 end subscript superscript 3 over 2 end superscript open vertical bar straight x space sinπ space straight x close vertical bar space dx space equals space 3 over straight pi plus 1 over straight pi squared$

152 Views

Short Answer Type

193.
Show that:
$integral subscript 0 superscript 2 straight x square root of 2 minus straight x end root space equals fraction numerator 16 square root of 2 over denominator 15 end fraction$

Let I = $integral subscript 0 superscript 2 straight x square root of 2 minus straight x end root dx$
$equals space integral subscript 0 superscript 2 left parenthesis 2 minus straight x right parenthesis space square root of 2 minus left parenthesis 2 minus straight x right parenthesis space dx end root$ $open square brackets because space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets$
$equals space integral subscript 0 superscript 2 left parenthesis 2 minus straight x right parenthesis space square root of straight x dx space equals integral subscript 0 superscript 2 open parentheses 2 straight x to the power of 1 half end exponent minus straight x to the power of 3 over 2 end exponent close parentheses dx space equals space open square brackets 2 space fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction minus fraction numerator straight x to the power of begin display style 5 over 2 end style end exponent over denominator begin display style 5 over 2 end style end fraction close square brackets subscript 0 superscript 2 space equals space open square brackets 4 over 3 straight x to the power of 3 over 2 end exponent minus 2 over 5 straight x to the power of 5 over 2 end exponent close square brackets subscript 0 superscript 2 equals space open parentheses 4 over 3 2 to the power of 3 over 2 end exponent minus 2 over 5 2 to the power of 5 over 2 end exponent close parentheses minus left parenthesis 0 minus 0 right parenthesis space equals 4 over 3 cross times 2 square root of 2 minus 2 over 5 cross times 4 square root of 2 space equals 8 over 3 square root of 2 minus 8 over 5 square root of 2 equals space fraction numerator 40 square root of 2 minus 24 square root of 2 over denominator 15 end fraction space equals space fraction numerator 16 square root of 2 over denominator 15 end fraction$

147 Views

194.

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 0 superscript 1 space straight x space left parenthesis 1 minus straight x right parenthesis to the power of straight n space dx$

108 Views

195.

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sinx minus cosx over denominator 1 plus sinxcosx end fraction dx$

134 Views

196.

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript 2 straight x space log space tanx space dx space equals space 0$

110 Views

197.

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 0 superscript straight pi over 2 end superscript left parenthesis 2 space log space sinx space minus space log space sin space 2 straight x right parenthesis space dx$

410 Views

198.

Show that:
$integral subscript 0 superscript straight pi fraction numerator straight x space tanx over denominator secx space plus space cosx end fraction dx space equals space straight pi squared over 4$

100 Views

199.

Show that:
$integral subscript 0 superscript infinity log space open parentheses straight x plus 1 over straight x close parentheses. space fraction numerator dx over denominator 1 plus straight x squared end fraction space equals space straight pi space log space 2$

132 Views

200.

Show that:
$integral subscript 0 superscript 1 fraction numerator log space left parenthesis 1 plus straight x right parenthesis over denominator 1 plus straight x squared end fraction space equals space straight pi over 8 space log space 2$