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

Let
$straight I space equals space integral subscript 1 superscript 2 open parentheses 1 over straight x minus fraction numerator 1 over denominator 2 straight x squared end fraction close parentheses space straight e to the power of 2 straight x end exponent space dx$
$equals space integral subscript 1 superscript 2 1 over straight x straight e to the power of 2 straight x end exponent dx space plus space integral fraction numerator negative 1 over denominator 2 straight x squared end fraction straight e to the power of 2 straight x end exponent dx$
$equals space open square brackets 1 over straight x straight e to the power of 2 straight x end exponent over 2 close square brackets subscript 1 superscript 2 space minus space integral subscript 1 superscript 2 fraction numerator negative 1 over denominator straight x squared end fraction straight e to the power of 2 straight x end exponent over 2 dx plus integral subscript 1 superscript 2 fraction numerator negative 1 over denominator 2 straight x squared end fraction straight e to the power of 2 straight x end exponent dx$ [integrating by parts]
$equals space 1 half open square brackets straight e to the power of 2 straight x end exponent over straight x close square brackets subscript 1 superscript 2 space minus space integral subscript 1 superscript 2 fraction numerator negative 1 over denominator 2 straight x squared end fraction straight e to the power of 2 straight x end exponent dx plus integral subscript 1 superscript 2 fraction numerator negative 1 over denominator 2 space straight x squared end fraction straight e to the power of 2 straight x end exponent dx$
$equals space 1 half open square brackets straight e to the power of 4 over 2 minus straight e squared over 1 close square brackets space equals space 1 fourth left parenthesis straight e to the power of 4 minus 2 straight e squared right parenthesis space equals 1 fourth straight e squared left parenthesis straight e squared minus 2 right parenthesis$

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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

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

138 Views