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

• Mathmanman
In summary, the integral of dx/[√sinx + √cosx]^4 from 0 to ∏/2 can be solved using a u-substitution and simplifying the resulting expression. However, when evaluated numerically, the answer is found to be exactly 1/3. There may be an easier method to solve this integral, but it has not been found yet.
Mathmanman

## Homework Statement

∫ [0,∏/2] dx/[√sinx + √cosx]^4

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.

Mathmanman said:

## Homework Statement

∫ [0,∏/2] dx/[√sinx + √cosx]^4

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.

Mathmanman said:

## Homework Statement

∫ [0,∏/2] dx/[√sinx + √cosx]^4

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.

Ray Vickson said:
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?

gopher_p said:
Any chance you tried a u-sub?

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

Ray Vickson said:
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.

Dick said:
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.

Ray Vickson said:
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??

## What is the process for evaluating the given integral?

The process for evaluating the given integral is to first simplify the expression inside the integral by using trigonometric identities and properties. Then, use the substitution method or integration by parts to solve the integral.

## What are some tips for simplifying the expression inside the integral?

Some tips for simplifying the expression inside the integral include using the trigonometric identities for sine and cosine, factoring out common terms, and using the Pythagorean identity.

## What is the range of values for the variable x in the integral?

The range of values for the variable x in the integral is from 0 to π/2. This is indicated by the limits of integration [0,π/2].

## How can the given integral be solved without using substitution or integration by parts?

The given integral can be solved without using substitution or integration by parts by using a trigonometric substitution. Specifically, you can use the substitution x = π/2 - θ to transform the integral into a simpler form.

## What is the purpose of evaluating the given integral?

The purpose of evaluating the given integral is to find the value of the definite integral for the given range of values. This can be useful in solving various problems in physics, engineering, and other fields that involve calculating areas, volumes, or other quantities.

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