Evaluate the following 1.1. \( \int\left(x^{3}-3 e^{2}+\frac{\pi}{x}-5 \sin x\right) d x \) \( 1.2 \cdot \int \frac{x}{x^{3}+x} d x \) 1.3. \( \int x^{4}\left(3-x^{5}\right)^{7} d x \) 1.4. \( \int \frac{x^{3}}{\sqrt{x^{2}+1}} d x \) 1.5. \( \int \frac{1}{x^{2}+2 x+5} d x \) \( 1.6 . \int \frac{\tanh (3+4 x)}{} d x \)

The motion of a particle performing damped vibrations is given by \( \gamma=e^{-t} \operatorname{Sin} 2 t, \gamma \) being the displacement trom its mean position at time \( t \). Show that \( \gamma \) is a maximum when \( t=1 / 2 \tan ^{-1}(2) \) and determine this maximum displacement to three significant figures.

Determina expresau a. \( \int 8 x\left(4 x^{2}-3\right) \) b. \( \int 2 t^{-3}\left(4 t^{-2}\right)^{-} \) c. \( \int_{1}^{3}\left(8 x-2 x^{2}\right) \)

(v) The value of \( \int_{-\infty}^{\infty} \delta(t) d t \) is: (a) 0 (b) 1 (c) -1 (d) \( \infty \) (vi) Solution of \( \frac{d^{2} x}{d x^{2}}-k x=0 \), when \( k=0 \) is: (a) \( c_{1} x+c_{2} \) (b) \( c_{1} \cos x+c_{2} \sin x \) (c) \( c_{1} e^{\lambda x}+c_{2} e^{-\lambda x} \) (d) None of these (vii) Equation \( u_{t u}-c^{2} u_{x z}=0 \) is: (a) one dimensional wave equation P.T.O.

The minimum value of the function \( f(x)=\frac{x}{x+1} \) is

\( \int ( 2 x + 3 ) ^ { 2 } \sqrt[ 3 ] { x - 2 } \)