grandes-ecoles 2016 QI.A.4

grandes-ecoles · France · centrale-maths2__mp Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Show $$\left\{ \begin{array}{l} \forall x > 0, \quad f(x) = x \ln(x) - \frac{1}{2} x \ln\left(x^{2} + 1\right) - \arctan(x) + \frac{\pi}{2} \\ f(0) = \frac{\pi}{2} \end{array} \right.$$

For $x \in \mathbb{R}^{+}$, we define
$$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$
Show
$$\left\{ \begin{array}{l} \forall x > 0, \quad f(x) = x \ln(x) - \frac{1}{2} x \ln\left(x^{2} + 1\right) - \arctan(x) + \frac{\pi}{2} \\ f(0) = \frac{\pi}{2} \end{array} \right.$$

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