grandes-ecoles 2016 QI.A.3

grandes-ecoles · France · centrale-maths2__mp Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Express $f^{\prime\prime}$ on $]0, +\infty[$ using standard functions and deduce that $$\forall x > 0, \quad f^{\prime}(x) = \ln(x) - \frac{1}{2} \ln\left(x^{2} + 1\right)$$

For $x \in \mathbb{R}^{+}$, we define
$$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$
Express $f^{\prime\prime}$ on $]0, +\infty[$ using standard functions and deduce that
$$\forall x > 0, \quad f^{\prime}(x) = \ln(x) - \frac{1}{2} \ln\left(x^{2} + 1\right)$$

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2