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

151.

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

90 Views

152.

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin cubed straight x over denominator sin cubed straight x plus cos cubed straight x end fraction dx space equals space straight pi over 4$

106 Views

153.

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript space fraction numerator sin to the power of 4 straight x over denominator sin to the power of 4 straight x plus cos to the power of 4 straight x end fraction dx space equals space straight pi over 4$

94 Views

154.

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator cos to the power of 5 straight x over denominator sin to the power of 5 straight x plus cos to the power of 5 straight x end fraction dx space equals space straight pi over 4$

89 Views

155.
Evaluate:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin to the power of begin display style 3 over 2 end style end exponent straight x over denominator sin to the power of begin display style 3 over 2 end style end exponent straight x plus cos to the power of begin display style 3 over 2 end style end exponent straight x end fraction dx space$

Let I = $integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin to the power of 3 divided by 2 end exponent straight x over denominator sin to the power of 3 divided by 2 end exponent straight x plus cos to the power of 3 divided by 2 end exponent straight x end fraction dx$ ...(1)
Then $straight I space equals integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin to the power of 3 divided by 2 end exponent open parentheses begin display style straight pi over 2 end style minus straight x close parentheses over denominator sin to the power of 3 divided by 2 end exponent open parentheses begin display style straight pi over 2 end style minus straight x close parentheses plus cos to the power of 3 divided by 2 end exponent open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end fraction dx$ $open square brackets because space space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space space equals integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets$
$therefore space space space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator cos to the power of 3 divided by 2 end exponent straight x over denominator cos to the power of 3 divided by 2 end exponent straight x plus sin to the power of 3 divided by 2 end exponent straight x end fraction dx.$ ...(2)
Adding (1) and (2), we get,
$2 space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript open square brackets fraction numerator sin to the power of 3 divided by 2 end exponent straight x over denominator sin to the power of 3 divided by 2 end exponent straight x plus cos to the power of 3 divided by 2 end exponent straight x end fraction plus fraction numerator cos to the power of 3 divided by 2 end exponent straight x over denominator cos to the power of 3 divided by 2 end exponent straight x plus sin to the power of 3 divided by 2 end exponent straight x end fraction close square brackets dx$

$equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin to the power of 3 divided by 2 end exponent straight x plus cos to the power of 3 divided by 2 end exponent straight x over denominator sin to the power of 3 divided by 2 end exponent straight x plus cos to the power of 3 divided by 2 end exponent straight x end fraction dx space equals space integral subscript 0 superscript straight pi divided by 2 end superscript 1 space dx space equals space open square brackets straight x close square brackets subscript 0 superscript straight pi divided by 2 end superscript space equals space straight pi over 2 minus 0 space equals space straight pi over 2$

$therefore space space space 2 space straight I space equals space straight pi over 2 space space space space space space space rightwards double arrow space space space space space space space straight I space equals space straight pi over 4$

108 Views

156.

Evaluate:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin to the power of begin display style 5 over 3 end style end exponent straight x over denominator sin to the power of begin display style 5 over 3 end style end exponent straight x plus cos to the power of begin display style 5 over 3 end style end exponent straight x end fraction dx space equals space straight pi over 4$

94 Views

157.

Evaluate:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator square root of cotx over denominator square root of tanx plus square root of cotx end fraction dx space equals space straight pi over 4$

106 Views

158.

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 1 plus cot space straight x end fraction space equals space straight pi over 4$

94 Views

Long Answer Type

159.

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 1 plus square root of tan space straight x end root end fraction space equals space straight pi over 4$

78 Views

Short Answer Type

160.

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 1 plus square root of cot space straight x end root end fraction space equals space straight pi over 4$

81 Views

Previous Year Papers

Test Series

Test Yourself

Short Answer Type

155. Evaluate:

Long Answer Type

Short Answer Type