Question

Evaluate:
$integral fraction numerator straight x squared over denominator left parenthesis straight x squared plus 4 right parenthesis left parenthesis straight x squared plus 9 right parenthesis end fraction dx$

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Solution

Let space straight x squared equals straight y fraction numerator straight x squared over denominator open parentheses straight x squared plus 4 close parentheses space left parenthesis straight x squared plus 9 right parenthesis end fraction space equals space fraction numerator straight y over denominator left parenthesis straight y plus 4 right parenthesis left parenthesis straight y plus 9 right parenthesis end fraction equals fraction numerator straight A over denominator straight y plus 4 end fraction plus fraction numerator straight B over denominator straight y plus 9 end fraction straight y space equals space straight A left parenthesis straight y plus 9 right parenthesis plus straight B left parenthesis straight y plus 4 right parenthesis Comparing space both space sides comma straight A plus straight B equals 1 space and space 9 straight A plus 4 straight B space equals space 0 Solving comma space We space get space straight A equals fraction numerator negative 4 over denominator 5 end fraction space and space straight B space equals 9 over 5 therefore space space straight I space equals space integral open square brackets fraction numerator negative 4 over denominator 5 left parenthesis straight x squared plus 4 right parenthesis end fraction plus fraction numerator 9 over denominator 5 left parenthesis straight x squared plus 9 right parenthesis end fraction close square brackets dx space space space space space equals negative 4 over 5 cross times 1 half tan to the power of negative 1 end exponent open parentheses straight x over 2 close parentheses plus 9 over 5 cross times 1 third tan to the power of negative 1 end exponent open parentheses straight x over 3 close parentheses plus straight C space space space space space equals negative 2 over 5 tan to the power of negative 1 end exponent straight x over 2 plus 3 over 5 tan to the power of negative 1 end exponent straight x over 3 plus straight C

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Integrals

Evaluate:
$integral fraction numerator straight x squared over denominator left parenthesis straight x squared plus 4 right parenthesis left parenthesis straight x squared plus 9 right parenthesis end fraction dx$

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