Answer
(a) Since
on (0,1] and
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_102.png)
converges, so does
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_103.png)
(b) Since
on [1,∞) and
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_105.png)
diverges, so does
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_106.png)
(c) Since
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_107.png)
and
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_108.png)
diverges, so does the original integral
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_109.png)
(d) We can't tell what this function does. The function g(x) is less than
on [0,1), but the integral
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_112.png)
diverges, which doesn't tell us anything. The function g(x) is greater than
on [0,1), but
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_114.png)
converges, so this doesn't tell us anything either!
(e) We can't tell what
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_115.png)
does. Since
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_116.png)
converges, it doesn't help to know that
on [1,∞).
Since
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_118.png)
diverges, it doesn't help to know that
on [1,∞).
(f) We can't tell what
![](https://media1.shmoop.com/images/calculus/calc_indefint_compimpropint_latek_120.png)
does, because we can't tell if either integral on the right-hand side diverges.