## HW SIX

1. (6.5)

(a)

Let us calculate the value of this intergral numerically using the adaptive quadrature with machine precision. The result is

Almost same result is obtained by evaluating analytically

We therefore conclude that it is likely that the statement that is true.

Note that this calculation does not provide a proof that this is true, only a demonstration that this is plausible assumption. Differnce in the last few digits are to be expected due to the complexity of this integral.

(b)

Similarly, ccccx

while

so we conclude that , so that this statement is also likely to be true.

(c)

while

so we confidently conclude that and this statement is false.

(d)

which is close to

So this statement is again likely to be true.

(e)

while

so we confidently conclude that .

We emphasize that using numerical methods to analyze analytic formulas of this kind can confidently proove that , but can only give indications that , with out prooving it.

2. (6.12) To evaluate the area under the curve

we need to evaluate numerically the integral