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

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165.
Evaluate $integral subscript negative straight pi over 4 end subscript superscript straight pi over 4 end superscript space sin squared straight x space dx$

Let I = $integral subscript straight pi over 4 end subscript superscript straight pi over 4 end superscript sin squared straight x space dx space equals space 2 integral subscript 0 superscript straight pi over 4 end superscript sin squared straight x space dx$
[∵ sin² x is an even function as sin² (–x) = sin² x]
$equals space integral subscript 0 superscript straight pi over 4 end superscript space space 2 space sin squared straight x space dx space equals space integral subscript 0 superscript straight pi over 4 end superscript left parenthesis 1 minus cos space 2 straight x right parenthesis space dx space equals space open square brackets straight x minus fraction numerator sin space 2 straight x over denominator 2 end fraction close square brackets subscript 0 superscript straight pi over 4 end superscript equals space open parentheses straight pi over 4 minus 1 half sin straight pi over 2 close parentheses space minus space open parentheses 0 minus 1 half sin space 0 close parentheses space equals space open parentheses straight pi over 4 minus 1 half close parentheses space minus space space left parenthesis 0 space minus space 0 right parenthesis space equals space straight pi over 4 minus 1 half$

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166.

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

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167.

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

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168.

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

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