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

92 Views

163.

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

97 Views

Short Answer Type

164.
Evaluate $integral subscript 0 superscript straight pi space sin squared straight x space cos cubed straight x space dx.$

Let I = $integral subscript 0 superscript straight pi sin squared straight x space cos cubed straight x space dx space equals space integral subscript 0 superscript straight pi straight f left parenthesis straight x right parenthesis space dx$
where f(x) = $sin squared straight x space cos cubed straight x$
Now $straight f left parenthesis straight pi minus straight x right parenthesis space equals space sin squared left parenthesis straight pi minus straight x right parenthesis space cos cubed left parenthesis straight pi minus straight x right parenthesis space equals space minus sin squared straight x space cos cubed straight x$
$therefore space space space straight f left parenthesis straight pi minus straight x right parenthesis space space equals negative straight f left parenthesis straight x right parenthesis$
$therefore space space space integral subscript 0 superscript straight pi sin squared straight x space cosx space dx space equals space 0 space space space space space open square brackets because space space if space straight f left parenthesis 2 straight a minus straight x right parenthesis space equals space minus straight f left parenthesis straight x right parenthesis comma space then space integral subscript straight a superscript 2 straight a end superscript straight f left parenthesis straight x right parenthesis space dx space equals space 0 close square brackets$

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165.

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

105 Views

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
$integral subscript negative straight pi over 2 end subscript superscript straight pi over 2 end superscript cosx space dx$

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

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170.

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