Evaluate $$\int {\left( {{e^{\log x}} + \sin x} \right)\cos x\,\,dx.} $$

The value of the definite integral $$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx$$

Let $$a, b, c$$ be non-zero real numbers such that
$$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} } $$
Then the quadratic equation $$a{x^2} + bx + c = 0$$ has

A gas bubble, from an explosion under water, oscillates with a period T proportional to p^ad^bE^c. Where 'P' is the static pressure, 'd' is the density of water and 'E' is the total energy of the explosion. Find the value of a, b and c.