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Evaluate : ∫(1+logx)/(x(2+logx)(3+logx))dx - HSC Arts 12th Board Exam - Mathematics and Statistics

ConceptEvaluation of Definite Integrals by Substitution

Question

Evaluate:  int (1+logx)/(x(2+logx)(3+logx))dx

Solution

int(1+logx)/(x(2+logx)(3+logx))dx

Substitute logx = t..................(1)

therefore 1/xdx=dt

Hence, the integral becomes

int(1+t)/((2+t)(3+t))dt

=int(2+t-1)/((2+t)(3+t))dt

=int(2+t)/((2+t)(3+t))dt-int1/((2+t)(3+t))dt

=int1/(3+t)dt-int((t+3)-(t+2))/((2+t)(3+t))dt

=int1/(3+t)dt-[int(t+3)/((2+t)(3+t))dt-int(t+2)/((2+t)(3+t))dt]

=int1/(3+t)dt-int1/(2+t)dt+int1/(3+t)dt

=2int1/(3+t)dt-int1/(2+t)dt

=2int1/(3+t)dt-int1/(2+t)dt

Substituting the value of 't' from (1), we get

int(1+logx)/(x(2+logx)(3+logx))dx

2ln (3+ logx )-ln( 2+ logx)+ C

=log|(3+logx)^2/(2 + logx)| + C

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APPEARS IN

2015-2016 (March) (with solutions)
Question 6.1.2 | 3.00 marks

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Solution Evaluate : ∫(1+logx)/(x(2+logx)(3+logx))dx Concept: Evaluation of Definite Integrals by Substitution.
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