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Integrals

Short Answer Type

161.

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

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Long Answer Type

162.

Evaluate $integral subscript straight pi over 6 end subscript superscript straight pi over 3 end superscript fraction numerator dx over denominator 1 plus square root of tanx end fraction$

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163.

Evaluate $integral subscript straight pi over 6 end subscript superscript straight pi over 3 end superscript fraction numerator square root of sin space straight x end root over denominator square root of sin space straight x end root plus square root of cos space straight x end root end fraction dx$

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Short Answer Type

164.

Evaluate $integral subscript 0 superscript straight pi space sin squared straight x space cos cubed straight x space dx.$

115 Views

165.

Evaluate $integral subscript negative straight pi over 4 end subscript superscript straight pi over 4 end superscript space sin squared straight x space dx$

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166.

Evaluate $integral subscript negative straight pi over 4 end subscript superscript straight pi over 4 end superscript cos squared xdx$

116 Views

167.
By using the properties of definite integrals, evaluate
$integral subscript negative straight pi over 2 end subscript superscript straight pi over 2 end superscript cosx space dx$

Let
$straight I space equals space integral subscript negative straight pi over 2 end subscript superscript straight pi over 2 end superscript cosx space dx space equals space 2 integral subscript 0 superscript straight pi over 2 end superscript space cosx space dx$
[∴ cos x is an even function as cos (–x) = cos x]
$equals space 2 open square brackets sin space straight x close square brackets subscript 0 superscript straight pi divided by 2 end superscript space equals space 2 open parentheses sin straight pi over 2 minus sin space 0 close parentheses space equals space 2 space left parenthesis 1 minus 0 right parenthesis space equals space 2.$

98 Views

168.

Evaluate $integral subscript negative 1 end subscript superscript 1 sin to the power of 5 straight x space cos to the power of 4 straight x space dx$

108 Views

169.

Evaluate $integral subscript negative 1 end subscript superscript 1 space straight x to the power of 17 space cos to the power of 4 straight x space dx.$

85 Views

170.

Evaluate $integral subscript negative 1 end subscript superscript 1 space log space open parentheses fraction numerator 2 plus straight x over denominator 2 minus straight x end fraction close parentheses dx.$