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Integrals

Question

Wired Faculty App

Evaluate:
$integral fraction numerator 5 straight x minus 2 over denominator left parenthesis 1 plus 2 straight x plus 3 straight x squared end fraction dx$

Solution

integral fraction numerator 5 straight x minus 2 over denominator 1 plus 2 straight x plus 3 straight x squared end fraction dx equals 5 integral fraction numerator straight x minus begin display style 2 over 5 end style over denominator 1 plus 2 straight x plus 3 straight x squared end fraction dx equals 5 over 6 integral fraction numerator 6 straight x minus begin display style 12 over 5 end style over denominator 1 plus 2 straight x plus 3 straight x squared end fraction dx equals 5 over 6 integral fraction numerator 6 straight x plus 2 minus begin display style 12 over 5 end style minus 2 over denominator 1 plus 2 straight x plus 3 straight x squared end fraction dx equals 5 over 6 integral fraction numerator 6 straight x plus 2 minus begin display style 22 over 5 end style over denominator 1 plus 2 straight x plus 3 straight x squared end fraction dx equals 5 over 6 integral fraction numerator 6 straight x plus 2 over denominator 1 plus 2 straight x plus 3 straight x squared end fraction dx minus 5 over 6 cross times 22 over 5 integral fraction numerator 1 over denominator 3 open curly brackets open parentheses straight x plus begin display style 1 third end style close parentheses squared plus begin display style 2 over 9 end style close curly brackets end fraction dx

equals 5 over 6 log open vertical bar 1 plus 2 straight x plus 3 straight x squared close vertical bar minus 11 over 9 integral fraction numerator 1 over denominator open parentheses straight x plus begin display style 1 third end style close parentheses squared plus begin display style 2 over 9 end style end fraction dx equals 5 over 6 log open vertical bar 1 plus 2 straight x plus 3 straight x squared close vertical bar minus 11 over 9 cross times fraction numerator 3 over denominator square root of 2 end fraction tan to the power of negative 1 end exponent fraction numerator open parentheses straight x plus begin display style 1 third end style close parentheses over denominator begin display style fraction numerator square root of 2 over denominator 3 end fraction end style end fraction plus straight C equals 5 over 6 log open vertical bar 1 plus 2 straight x plus 3 straight x squared close vertical bar minus fraction numerator 11 over denominator 3 square root of 2 end fraction cross times tan to the power of negative 1 end exponent open parentheses fraction numerator 3 straight x plus 1 over denominator square root of 2 end fraction close parentheses plus straight C

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