\( ( \operatorname { Ln } ( 1 + x ) ) ^ { \prime \prime } = \)

Calcula el área entre las curvas \( y = x^3 - 3x \) y \( y = x + 1 \) sobre el intervalo definido por sus intersecciones.

(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

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 \( \forall x>0: \) \( \frac{1}{1+x}<\ln \left(1+\frac{1}{x}\right)<\frac{1}{x} \)

Choose the correct answer from those givens: (1) If \( \operatorname{Lim}_{x \rightarrow 0} \frac{\cos x-\cos ^{a} x}{x^{2}}=1 \), then \( a=\ldots \ldots \ldots \ldots \ldots \) \( \begin{array}{llll}\text { (a) } 1 & \text { (b) zero } & \text { (c) } 2 & \text { (d) } 3\end{array} \)

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?