Mathematical Tools for Physics - Department of Physics - University

1—Basic Stuff 18

20 Given that

∫∞

−∞dx/(1 +x

(^2) ) =π,i.e.you don’t have to derive this, what then is∫∞

−∞dx/(α+x

(^2) )?
Now differentiate the result and find the two integrals

∫∞

−∞dx/(1 +x

(^2) ) (^2) and∫∞

−∞dx/(1 +x

(^2) ) (^3).
21 Derive the product rule as a special case of Eq. (1.20).
22 The third paragraph of section1.6has two simple equations in arithmetic. What common identities
about the integral do these correspond to?

23 Plot a graph ofy=exwithyandxin meters (xhorizontal andyvertical). Start at the origin

and walk along thex-axis at one meter per second. When you are at the 20-meter point, where is the

ycoordinate and how fast is it rising? Not just numbers: compare both to real things.