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Integrals

Question
CBSEENMA12035639
Wired Faculty App

Find space colon integral fraction numerator straight x squared plus straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction dx

Solution
Taking space the space given space function straight I equals integral fraction numerator straight x squared plus straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction dx Let space fraction numerator straight x squared space plus space straight x space plus space 1 over denominator left parenthesis straight x squared space plus space 1 right parenthesis space left parenthesis straight x space plus space 2 right parenthesis end fraction space space equals space fraction numerator straight A over denominator straight x space plus space 2 end fraction space plus space fraction numerator Bx space plus space straight C over denominator straight x squared plus 1 end fraction equals space fraction numerator straight A left parenthesis straight x squared space plus space 1 right parenthesis space plus space left parenthesis Bx space space plus space straight C right parenthesis left parenthesis straight x space plus space 2 right parenthesis over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction equals fraction numerator left parenthesis straight A plus straight B right parenthesis straight x squared plus left parenthesis 2 straight B plus straight C right parenthesis space straight x plus left parenthesis 2 straight C plus straight A right parenthesis over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction Thus space equating space the space coefficients comma space we space have comma straight A space plus space straight B space equals 1 space.. space left parenthesis straight i right parenthesis 2 straight B space space plus space straight C space equals space 1 space.. left parenthesis ii right parenthesis 2 straight C space plus space straight A space equals space 1 space.. space left parenthesis iii right parenthesis solving space the space above space three space equations comma space we space have comma straight A space equals 3 over 5 comma space straight B space equals 2 over 5 space and space straight C equals 1 fifth therefore comma fraction numerator straight x squared plus straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction space fraction numerator straight A over denominator straight x plus 2 end fraction plus fraction numerator Bx plus straight C over denominator straight x squared plus 1 end fraction rightwards double arrow space fraction numerator straight x squared plus straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction space equals space fraction numerator 3 over denominator 5 left parenthesis straight x plus 2 right parenthesis end fraction space plus space fraction numerator 2 straight x plus 1 over denominator 5 left parenthesis straight x squared plus 1 right parenthesis end fraction therefore comma straight I equals integral fraction numerator straight x squared plus straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction dx equals integral open square brackets fraction numerator 3 over denominator 5 space left parenthesis straight x plus 2 right parenthesis end fraction plus fraction numerator 2 straight x plus 1 over denominator 5 left parenthesis straight x squared plus 1 right parenthesis end fraction dx close square brackets equals 3 over 5 integral fraction numerator dx over denominator left parenthesis straight x plus 2 right parenthesis end fraction dx space plus space 1 fifth integral fraction numerator 2 straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis end fraction dx equals 3 over 5 log space left parenthesis straight x plus 2 right parenthesis space plus 1 fifth integral fraction numerator 2 straight x over denominator left parenthesis straight x squared plus 1 right parenthesis end fraction dx space plus 1 fifth integral fraction numerator dx over denominator left parenthesis straight x squared plus 1 right parenthesis end fraction equals 3 over 5 log space left parenthesis straight x plus 2 right parenthesis space plus 1 fifth log space left parenthesis straight x squared plus 1 right parenthesis plus 1 fifth tan to the power of negative 1 end exponent space straight x space plus space straight C

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