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

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

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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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