Chapter 5.5, Problem 95E

 (a).

To determine

Whether the statement ${\displaystyle \int f\left(x\right)f^{\prime }\left(x\right)dx=\frac{1}{2}{\left(f\left(x\right)\right)}^2+C},\text{where }f,f^{\prime },f^{″}$ are continuous functions for all numbers”, is true or false; give an explanation or counter example.

(b).

To determine

Whether the statement, “ ${\displaystyle \int {\left(f\left(x\right)\right)}^n{f}^{\prime }\left(x\right)dx=\frac{1}{n+1}{\left(f\left(x\right)\right)}^{n+1}+C},n\ne -1,\text{where }f,f^{\prime },f^{″}$ are continuous functions for all numbers”, is true or false with an explanation or counter example.

(c).

To determine

Whether the statement, “ ${\displaystyle \int \sin 2xdx}=2{\displaystyle \int \sin xdx}$” true or false; give an explanation or counter example.

(d).

To determine

Whether the statement, “ ${\displaystyle \int {\left(x^2+1\right)}^{9}dx}=\frac{{\left(x^2+1\right)}^{10}}{10}+C$” is true or false; give an explanation or counter example.

(e).

To determine

Whether the statement, “ ${\displaystyle \underset{a}{\overset{b}{\int }}{f}^{\prime }\left(x\right)f^{″}\left(x\right)dx}=f^{\prime }\left(b\right)-f^{\prime }\left(a\right)$” is true or false; give an explanation or counter example.