Chapter 8.2, Problem 77E

a.

To determine

To show: ${\displaystyle \int f\left(x\right)g\left(x\right)}dx=f\left(x\right)G_1\left(x\right)-{\displaystyle \int f^{\prime }\left(x\right)}{G}_1\left(x\right)dx$

b.

To determine

To show: ${\displaystyle \int f\left(x\right)g\left(x\right)}dx=f\left(x\right)G_1\left(x\right)-f^{\prime }\left(x\right)G_2\left(x\right)+{\displaystyle \int f^{″}}\left(x\right)G_2\left(x\right)dx$

c.

To determine

To write: The integral formula that results from three applications of integration by parts, and construct the associated tabular integration table.

d.

To determine

To evaluate: The integral ${\displaystyle \int x^2{e}^{0.5x}}dx$ and find the required number of the row required in the table.

e.

To determine

To evaluate: The integral ${\displaystyle \int x^3\cos x}dx$ by constructing an appropriate table: Explain the reason behind that the process terminates after four rows of the table have been filled in.

f

To determine

To explain: The tabular integration is particularly suited to integrals of the form ${\displaystyle \int p_n\left(x\right)g\left(x\right)}dx$.