Exams Joint Entrance Examination Mathematics Indefinite Integrals Question 184 Question 184 of 416 Easy MCQ
The value of the integral ∫ − 1 1 log ( x + x 2 + 1 ) d x \displaystyle\int_{-1}^1\log\left(x+\sqrt{x^2+1}\right)dx ∫ − 1 1 log ( x + x 2 + 1 ) d x is:
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The integrand log ( x + x 2 + 1 ) = sinh − 1 x \log\left(x+\sqrt{x^2+1}\right)=\sinh^{-1}x log ( x + x 2 + 1 ) = sinh − 1 x is an odd function (since replacing x → − x x\to-x x → − x gives log ( − x + x 2 + 1 ) = − log ( x + x 2 + 1 ) \log\left(-x+\sqrt{x^2+1}\right)=-\log\left(x+\sqrt{x^2+1}\right) log ( − x + x 2 + 1 ) = − log ( x + x 2 + 1 ) ).
Since the limits are symmetric about 0 0 0 , I = 0 I=0 I = 0 . Answer the question to unlock