Integral arctg

[Loki123]

Calculate the integral written at the top of the page.
Two question:
How do I use 1, 0 for xarctgx? All at once or seperatly? I wrote the two ways, tell me which is correct, if any.
How can I get the correct answer:

 

[BigBeachBanana]

They are equivalent.
[math]\frac{1}{2}\ln(2)=\ln(2)^{\frac{1}{2}}=\ln(\sqrt{2})[/math]

 

[Loki123]

They are equivalent.
[math]\frac{1}{2}\ln(2)=\ln(2)^{\frac{1}{2}}=\ln(\sqrt{2})[/math]

Ooh, completely missed that. What about question 1?

 

[BigBeachBanana]

Ooh, completely missed that. What about question 1?

Are you referring to this?
[math]x\arctan(x)\biggr\rvert_{0}^{1}=\left[1\cdot \arctan(1)\right]-\cancel{\left[0\cdot \arctan(0)\right]}=\frac{\pi}{4}[/math]The first way is wrong. You got lucky that they evaluate to the same value.

 

[Dr.Peterson]

What about question 1?

This is wrong:

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You have to evaluate the entire expression for each limit.

 

[Loki123]

Are you referring to this?
[math]x\arctan(x)\biggr\rvert_{0}^{1}=\left[1\cdot \arctan(1)\right]-\cancel{\left[0\cdot \arctan(0)\right]}=\frac{\pi}{4}[/math]The first way is wrong. You got lucky that they evaluate to the same value.

Yes thank you!