7. .

Now using Integration by Parts with and and the integral becomes

.

Now make the substitution and the integral becomes

.

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- January 27th 2011, 12:31 AMProve It
7. .

Now using Integration by Parts with and and the integral becomes

.

Now make the substitution and the integral becomes

. - January 27th 2011, 12:36 AMProve It
8.

.

Now make the substitution and the integral becomes

. - January 27th 2011, 12:58 AMProve It
9.

after making the substitution

which I'm sure scrubs up much nicer than does here... - January 27th 2011, 01:28 AMProve It
12.

.

Now make the substitution and the integral becomes

.

Now using Partial Fractions:

and

.

So

. - January 27th 2011, 01:35 AMProve It
17. .

Now let and the integral becomes

.

Now using Partial Fractions:

and

and .

So

- January 27th 2011, 01:41 AMProve It
19. .

Let and the integral becomes

after using Partial Fractions

. - January 27th 2011, 01:43 AMTed
Prove It:

for #17, If you multiply and devide by it will be solved in 1 second. - January 27th 2011, 01:55 AMProve It
21.

.

Note that is a semicircle centred at of radius .

So this integral is the area of that semicircle.

. - January 27th 2011, 02:12 AMProve It
.

Now let and .

When and when .

The integral becomes

. - January 27th 2011, 02:20 AMProve It
25.

.

Now let and the integral becomes

. - January 27th 2011, 02:26 AMProve It
23. .

Let and the integral becomes

.

Now let and the integral becomes

. - January 27th 2011, 02:46 AMProve It
10.

Let and the integral becomes

. - January 27th 2011, 02:53 AMProve It
Just out of interest, having never been in an Integral Bee before, how many do you have to get right in the time limit to qualify?

- January 27th 2011, 04:24 AMTheCoffeeMachine
Don't know, I'm wondering that too. It might depend on the boundaries of that particular year. I think with sufficient training and wise choice of questions, most people can get more than ten right within the time limit. But you have to keep in mind that they are not doing them in the regular way -- with the explicit substitutions and all. For example, it's much more efficient to do # 10 as follows:

- January 27th 2011, 04:47 AMTheCoffeeMachine
By the way, here is a head-to-head video of the final round (not sure which year).

It took the winner about 30 seconds to do . What do you think?