350. Another (easy) integrals.
by Virgil Nicula, Jul 18, 2012, 6:07 PM
PP1. Ascertain 
where 
and 
.
Proof.
where 
and 
. In conclusion, 
.
PP2. Ascertain
where 
.
Proof.
where 
and
.
In conclusion,
.
PP3. Ascertain
.
.
where 
.
Remark.
.
Prove similarly that
.
PP4. Ascertain![$\int x^p(x^p+a)\sqrt[p]{x^p+2a}\ \mathrm{dx}$](//latex.artofproblemsolving.com/1/0/0/1002f92c3aea46a4675db9495ab92ad1c1bf953d.png)
where 
and 
.
Proof.![$\int x^p(x^p+a)\sqrt[p]{x^p+2a}\ \mathrm{dx}=$](//latex.artofproblemsolving.com/a/1/d/a1db9779fe693bbaee1632a66d59835694453c34.png)
![$\frac {1}{2p}\cdot\int \left(2px^{2p-1}+2apx^{p-1}\right)\sqrt [p]{x^{2p}+2ax^p}\ \mathrm{dx}=$](//latex.artofproblemsolving.com/a/9/1/a91ae187d99cbee32bcffa1534f33d63cb28e114.png)
![$\boxed{\ \int x^p(x^p+a)\sqrt[p]{x^p+2a}\ \mathrm{dx}=\frac {1}{2(p+1)}x^p\left(x^{p}+2a\right)\sqrt [p]{x^p+2a}+\mathcal C\ }$](//latex.artofproblemsolving.com/c/a/1/ca19a61a399275e33e18c8e941d012e6cfae8f2e.png)
. Very nice exercise !
PP5. Ascertain
.
Proof.
. See 
from PP1.
Remark. This problem was given at one from the contests of O.J.M. Romania.
PP6. If
. Let 
, Where 
. Then find value of 
for which 
is smallest.
Proof. Note that
. Also, 
and 
for all 
. So, 
.
By AM-GM,
with equality iff 
. Thus, 
with equality iff .
PP7. Calculati
, unde 
.
Solutie.
.
, where 
.
Therefore,
.
Problema propusa. Determinati o relatie de recurenta pentru
, unde and 
.
where 
and 
.Proof.
where 
and 
. In conclusion, 
.PP2. Ascertain
where 
.Proof.
where 
and
.In conclusion,
.PP3. Ascertain
.
.
where 
.Remark.
.Prove similarly that
.PP4. Ascertain
![$\int x^p(x^p+a)\sqrt[p]{x^p+2a}\ \mathrm{dx}$](http://latex.artofproblemsolving.com/1/0/0/1002f92c3aea46a4675db9495ab92ad1c1bf953d.png)
where 
and 
.Proof.
![$\int x^p(x^p+a)\sqrt[p]{x^p+2a}\ \mathrm{dx}=$](http://latex.artofproblemsolving.com/a/1/d/a1db9779fe693bbaee1632a66d59835694453c34.png)
![$\frac {1}{2p}\cdot\int \left(2px^{2p-1}+2apx^{p-1}\right)\sqrt [p]{x^{2p}+2ax^p}\ \mathrm{dx}=$](http://latex.artofproblemsolving.com/a/9/1/a91ae187d99cbee32bcffa1534f33d63cb28e114.png)
![$\boxed{\ \int x^p(x^p+a)\sqrt[p]{x^p+2a}\ \mathrm{dx}=\frac {1}{2(p+1)}x^p\left(x^{p}+2a\right)\sqrt [p]{x^p+2a}+\mathcal C\ }$](http://latex.artofproblemsolving.com/c/a/1/ca19a61a399275e33e18c8e941d012e6cfae8f2e.png)
. Very nice exercise !PP5. Ascertain
.Proof.
. See 
from PP1.Remark. This problem was given at one from the contests of O.J.M. Romania.
PP6. If
. Let 
, Where 
. Then find value of 
for which 
is smallest.Proof. Note that
. Also, 
and 
for all 
. So, 
.By AM-GM,
with equality iff 
. Thus, 
with equality iff .PP7. Calculati
, unde 
. Solutie.
.
, where 
.Therefore,
.Problema propusa. Determinati o relatie de recurenta pentru
, unde and 
. This post has been edited 7 times. Last edited by Virgil Nicula, May 1, 2016, 6:34 PM
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