Evaluating Integral ∫ [0,∏/2] dx/[√sinx + √cosx]^4 | Simplify and Tips

[Mathmanman]

Homework Statement

∫ [0,∏/2] dx/[√sinx + √cosx]^4

Homework Equations

None

The Attempt at a Solution

∫ [0,∏/2] dx/[√sinx + √cosx]^4
= ∫ [0,∏/2] dx/[(√tanx + 1)^4 cosx^2]
= ∫ [0,∏/2] sec^2(x) dx/[(√tanx + 1)^4]

This is my attempt. I can only simplify. I'm stumped.
Can you also give me advice on solving integral problems like this?
Because it would be very nice to be able to evaluate problems like this without help.

 

[gopher_p]

Homework Statement

∫ [0,∏/2] dx/[√sinx + √cosx]^4

Homework Equations

None

The Attempt at a Solution

∫ [0,∏/2] dx/[√sinx + √cosx]^4
= ∫ [0,∏/2] dx/[(√tanx + 1)^4 cosx^2]
= ∫ [0,∏/2] sec^2(x) dx/[(√tanx + 1)^4]

This is my attempt. I can only simplify. I'm stumped.
Can you also give me advice on solving integral problems like this?
Because it would be very nice to be able to evaluate problems like this without help.

Try a u-sub.

 

[Ray Vickson]

Homework Statement

∫ [0,∏/2] dx/[√sinx + √cosx]^4

Homework Equations

None

The Attempt at a Solution

∫ [0,∏/2] dx/[√sinx + √cosx]^4
= ∫ [0,∏/2] dx/[(√tanx + 1)^4 cosx^2]
= ∫ [0,∏/2] sec^2(x) dx/[(√tanx + 1)^4]

This is my attempt. I can only simplify. I'm stumped.
Can you also give me advice on solving integral problems like this?
Because it would be very nice to be able to evaluate problems like this without help.

Maple manages to perform the indefinite integral, but the results are unenlightening: it is a 156-page expression involving complicated combinations of sin(x), cos(x), plus logarithms of such combinations, plus various Elliptic functions of such combinations. The definite integral is bordering on the un-doable, because of the need to evaluate that humongous expression numerically at the endpoints; just doing it numerically seems best. When letting Maple do it numerically (for increasingly many digits of accuracy) the given answer equals to the decimal representation of 1/3 to the given number of digits setting. So, it seems that the answer = 1/3 exactly! There absolutely must be an easy (or, at least, easier) way to see this, but so far I have not found one.

For 20-digits in computations, Maple gets the answer as 0.33333333333333333333, while for 50-digits it gets 0.33333333333333333333333333333333333333333333333333, etc.

 

[gopher_p]

Maple manages to perform the indefinite integral, but the results are unenlightening: it is a 156-page expression involving complicated combinations of sin(x), cos(x), plus logarithms of such combinations, plus various Elliptic functions of such combinations. The definite integral is bordering on the un-doable, because of the need to evaluate that humongous expression numerically at the endpoints; just doing it numerically seems best. When letting Maple do it numerically (for increasingly many digits of accuracy) the given answer equals to the decimal representation of 1/3 to the given number of digits setting. So, it seems that the answer = 1/3 exactly! There absolutely must be an easy (or, at least, easier) way to see this, but so far I have not found one.

For 20-digits in computations, Maple gets the answer as 0.33333333333333333333, while for 50-digits it gets 0.33333333333333333333333333333333333333333333333333, etc.

Any chance you tried a u-sub?

 

[Ray Vickson]

Any chance you tried a u-sub?

I don't know what Maple did to get the 156-page answer.

 

[Dick]

I don't know what Maple did to get the 156-page answer.

Probably didn't think to pull out the sec^2 like Mathmanman and do the u-sub gopher_p suggested. Found a long way around.

 

[Ray Vickson]

Probably didn't think to pull out the sec^2 like Mathmanman and do the u-sub gopher_p suggested. Found a long way around.

You should do an appropriate substitution if you think that a one-line result is better than a 156-page one.

 

[Dick]

You should do an appropriate substitution if you think that a one-line result is better than a 156-page one.

I did. u=tan(x). Isn't it better??