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

Solution

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

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Integrals

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

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