Evaluate the following integral:
$integral subscript 0 superscript 1 fraction numerator 5 straight x over denominator left parenthesis 4 plus straight x squared right parenthesis squared end fraction dx$

Solution

Let I = $integral subscript 0 superscript 1 fraction numerator 5 straight x over denominator left parenthesis 4 plus straight x squared right parenthesis squared end fraction dx space equals space 5 integral subscript 0 superscript 1 fraction numerator straight x over denominator left parenthesis 4 plus straight x squared right parenthesis squared end fraction dx$
Put $4 plus straight x squared space equals space straight y semicolon space space space space space therefore space 2 space straight x space dx space equals space dy space rightwards double arrow space space space straight x space dx space equals space 1 half dy$
When x = 0, y = 4 + 0 = 4
$therefore$ I = $5 over 2 integral subscript 4 superscript 5 1 over straight y squared dy space equals space 5 over 2 integral subscript 2 superscript 5 straight y to the power of negative 2 end exponent dy space equals space 5 over 2 open parentheses fraction numerator straight y to the power of negative 1 end exponent over denominator negative 1 end fraction close parentheses subscript 4 superscript 5$
$equals negative 5 over 2 open square brackets 1 over straight y close square brackets subscript 4 superscript 5 space equals space minus 5 over 2 open square brackets 1 fifth minus 1 fourth close square brackets space equals space minus 5 over 2 open square brackets fraction numerator 4 minus 5 over denominator 20 end fraction close square brackets equals negative 5 over 2 cross times fraction numerator negative 1 over denominator 20 end fraction equals space 1 over 8$

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Integrals

Evaluate the following integral:
$integral subscript 0 superscript 1 fraction numerator 5 straight x over denominator left parenthesis 4 plus straight x squared right parenthesis squared end fraction dx$

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