Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Integration, Exercise 2: EXERCISE 5A

For the following graph, Find the exact area of the shaded region.

Show that $\frac{d}{dx}\left(x{e}^x-e^x\right)=x{e}^x$.

For the following graph, Find the area of the shaded region.

Find ${\int }_0^a\left(4e^{-2x}+5e^{-x}\right)dx$, where $a$ is a positive constant.

Find the value of${\int }_0^{\infty }\left(4e^{-2x}+5e^{-x}\right)dx$.

The diagram shows the curve $y=2e^x+8e^{-x}-7$ and its minimum point $M$. Find the area of the shaded region.

The diagram shows the graph of $y=e^x$. The points $(\ln 2,2)\text{ and }(\ln 3,3)$ lie on the curve. Find the value of ${\int }_{\ln 2}^{\ln 3}{e}^{x}dx$.

The diagram shows the graph of $y=e^x$. The points $(\ln 2,2)\text{ and }(\ln 3,3)$ lie on the curve. Show that ${\int }_2^3\ln y dy=\ln \left(\frac{27}{4e}\right)$.