(vi) \( \mathrm{L}^{-1}\left\{\tan ^{-1} \frac{1}{\mathrm{p}}\right\}= \)

A solid is formed with triangular cross-sections where each triangle's base is defined by the equation \(y = x^2\) on interval \([0, 2]\). What is the volume of this solid?

(iv) In integral equation of the form \( \int_{0}^{1} \frac{F(u) d u}{(t-u)^{n}}=G(t) \) is called. (a) Exact equation (b) Convolution type (c) Integro-differential equation (d) Abel's equation

\( \int _{0}^{\frac{\pi }{2}}\frac{1}{2+\cos (x)} d x \)

\( \int _{0}^{\frac{\pi }{2}}\frac{1}{2+\cos (x)} d x \)

1) Sout \( x>0 \). En appliquant le thécorème des accroissement fines à la fonction \( \ln \) montrer que: \( \exists c \in] x, x+1\left[\frac{1}{c}=\ln \left(1+\frac{1}{x}\right)\right. \) 2) \( M q: \exists c \in] 0, x\left[\right. \) arctan \( (x)=\frac{x}{1+c^{2}} \) 3) En déduire que \( \forall x>0: \) \( \frac{1}{1+x}<\ln \left(1+\frac{1}{x}\right)<\frac{1}{x} \)