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Integrals

Short Answer Type

131. Evaluate the following:

integral subscript 0 superscript straight pi fraction numerator dx over denominator 1 plus sin space straight x end fraction

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132. Evaluate the following:

integral subscript 0 superscript 1 sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses dx

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133. Evaluate the following:

integral subscript 0 superscript 1 cos to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction close parentheses dx

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134. Evaluate the following:

integral subscript 0 superscript 1 tan to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 minus straight x squared end fraction close parentheses dx

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135. Evaluate the following:

integral subscript 0 superscript 2 straight pi end superscript straight e to the power of straight x space sin open parentheses straight pi over 4 plus straight x over 2 close parentheses dx.

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136. Evaluate the following:

integral subscript 1 superscript 2 open parentheses 1 over straight x minus fraction numerator 1 over denominator 2 straight x squared end fraction close parentheses space straight e to the power of 2 straight x end exponent dx

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137. Evaluate the following:
$integral subscript straight pi over 2 end subscript superscript straight pi straight e to the power of straight x open parentheses fraction numerator 1 minus sinx over denominator 1 minus cosx end fraction close parentheses dx$

Let
I = $integral subscript straight pi over 2 end subscript superscript straight pi space straight e to the power of straight x open parentheses fraction numerator 1 minus sinx over denominator 1 minus cosx end fraction close parentheses dx space equals space integral subscript straight pi over 2 end subscript superscript straight pi straight e to the power of straight x open square brackets fraction numerator 1 minus 2 sin begin display style straight x over 2 end style cos begin display style straight x over 2 end style over denominator 2 space sin squared begin display style straight x over 2 end style end fraction close square brackets dx$
$equals space integral subscript straight pi over 2 end subscript superscript straight pi straight e to the power of straight x open square brackets fraction numerator 1 over denominator 2 sin squared begin display style straight x over 2 end style end fraction minus fraction numerator 2 sin begin display style straight x over 2 end style cos begin display style straight x over 2 end style over denominator 2 sin squared begin display style straight x over 2 end style end fraction close square brackets space dx space equals space integral subscript straight pi over 2 end subscript superscript straight pi straight e to the power of straight x open square brackets 1 half cosec squared straight x over 2 minus cot straight x over 2 close square brackets dx$
$equals space integral subscript straight pi over 2 end subscript superscript straight pi over 2 end superscript straight e to the power of straight x open square brackets negative cot space straight x over 2 plus 1 half cosec squared straight x over 2 close square brackets dx space equals space open square brackets straight e to the power of straight x open parentheses negative cot space straight x over 2 close parentheses close square brackets subscript straight pi over 2 end subscript superscript straight pi$
$open square brackets because space space integral space straight e to the power of straight x left curly bracket straight f left parenthesis straight x right parenthesis space plus space straight f apostrophe left parenthesis straight x right parenthesis right curly bracket space dx space equals space straight e to the power of straight x space straight f left parenthesis straight x right parenthesis close square brackets$
$equals space minus open square brackets straight e to the power of straight x space cot space straight x over 2 close square brackets subscript straight pi over 2 end subscript superscript straight pi space equals space minus open square brackets straight e to the power of straight pi space cot straight pi over 2 minus straight e to the power of straight pi over 2 end exponent space cot straight pi over 4 close square brackets space equals negative open square brackets straight e to the power of straight pi cross times 0 minus straight e to the power of straight pi over 2 end exponent cross times 1 close square brackets space equals space straight e to the power of straight pi over 2 end exponent$

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138. Evaluate the following:

integral subscript 0 superscript 2 straight pi end superscript space straight e to the power of straight x over 2 end exponent space sin space open parentheses straight x over 2 plus straight pi over 4 close parentheses dx

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139.

Evaluate the following:
$integral subscript 0 superscript 2 straight pi end superscript straight e to the power of straight x cos open parentheses straight pi over 4 plus straight x over 2 close parentheses dx.$

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Multiple Choice Questions

140.

The value of the integral $integral subscript 1 third end subscript superscript 1 fraction numerator left parenthesis straight x minus straight x cubed right parenthesis to the power of begin display style 1 third end style end exponent over denominator straight x to the power of 4 end fraction dx$

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