Chapter 7.6, Problem 113E

a.

To determine

To explain: The shaded region in the given figure has area ${\int }_0^{\ln a}{e}^y$ .

b.

To determine

To prove: ${\int }_1^a\ln xdx=a\ln a-{\int }_0^{\ln a}{e}^{y}dy$

c.

To determine

To prove: ${\int }_1^a\ln xdx=a\ln a-a+1$ .

d.

To determine

To find: The antiderivative of $\ln (x)$ using the result of ${\displaystyle {\int }_0^{\ln a}{e}^{y}dy}$ .