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

Let I = $integral subscript 0 superscript 1 space sin to the power of negative 1 end exponent space open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses dx$
Put x = tan θ so that dx = sec² θ dθ When x = 0, tan θ = 0 ⇒ θ = 0
When x = 1, $tan space straight theta space equals space 1 space space space space space rightwards double arrow space space space straight theta space equals space straight pi over 4$
$therefore space space space space straight I space equals space integral subscript 0 superscript straight pi over 4 end superscript space sin to the power of negative 1 end exponent space open parentheses fraction numerator 2 space tanθ over denominator 1 plus tan squared straight theta end fraction close parentheses. space sec squared straight theta space dθ$
$equals space integral subscript 0 superscript straight pi over 4 end superscript space sin to the power of negative 1 end exponent left parenthesis sin space 2 straight theta right parenthesis. space sec squared straight theta space dθ space equals space integral subscript 0 superscript straight pi over 4 end superscript 2 straight theta. space sec squared straight theta space dθ space equals space 2 integral subscript 0 superscript straight pi over 4 end superscript straight theta space. sec squared space straight theta space dθ$
$equals space 2 open curly brackets open square brackets straight theta space tan space straight theta close square brackets subscript 1 superscript straight pi over 4 end superscript minus integral subscript 0 superscript straight pi over 4 end superscript 1. space tanθ space dθ close curly brackets space space equals space 2 open curly brackets open square brackets straight theta space tanθ close square brackets subscript 0 superscript straight pi over 4 end superscript plus open square brackets log space cosθ close square brackets subscript 0 superscript straight pi over 4 end superscript close curly brackets equals space 2 open curly brackets straight pi over 4 tan straight pi over 4 minus 0 plus log space cos straight pi over 4 minus log space cos 0 close curly brackets space equals space 2 open square brackets straight pi over 4 cross times 1 minus 0 plus log fraction numerator 1 over denominator square root of 2 end fraction minus log space 1 close square brackets equals space 2 open square brackets straight pi over 4 plus log space 1 space minus space log space square root of 2 close square brackets space equals space straight pi over 2 minus log space 2$

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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

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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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