2 marksMathematicsIntegration and its applicationsIntegration techniquesHardabout 3 min

IIT JAM 2022 Question 53: Integration and its applications

If $\displaystyle\int t\log\!\left(1 + \dfrac{2}{t}\right)dt = g(t)\left(\dfrac{t^2}{2} - 2\right) + f(t)\dfrac{t^2}{2} + Kt + C$ , where $C$ is an arbitrary constant, then $2K$ is _______ (in integer).

Answer and worked solution

Answer: 2 to 2

Write $\log(1 + 2/t) = \ln(t+2) - \ln t$ and integrate by parts with $v = t^2/2$ . This gives $\dfrac{t^2}{2}\left[\ln(t+2) - \ln t\right] + \int \dfrac{t}{t+2}\,dt$ .

Since $\int \dfrac{t}{t+2}\,dt = t - 2\ln(t+2) + C$ , the result is $\ln(t+2)\left(\dfrac{t^2}{2} - 2\right) - \ln(t)\dfrac{t^2}{2} + t + C$ . Matching to the given form, $g(t) = \ln(t+2)$ , $f(t) = -\ln t$ and $Kt = t$ , so $K = 1$ and $2K = 2$ .

Plain-text version

If ∫ tlog (1 + 2/t)dt = g(t)(t²/2 - 2) + f(t)t²/2 + Kt + C, where C is an arbitrary constant, then 2K is _ (in integer).

Also written as: If integral tlog (1 + 2/t)dt = g(t)(t^2/2 - 2) + f(t)t^2/2 + Kt + C, where C is an arbitrary constant, then 2K is _ (in integer).

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What this question tests

This question is on Integration techniques, part of Integration and its applications in Mathematics.

Substitution, integration by parts, partial fractions and improper integrals.

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IIT JAM 2021 · Q44The value of the integral ∫₀⁹ (x - 1)/(1 + √x) dx is _ (in integer).

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