# Integrate (sinx+cosx)/sqrt(1+sin2x)

• Tanishq Nandan
In summary, the problem is that I substituted sin2x for a variable in every term, but now I can't express the whole term in terms of a simple integral.
Tanishq Nandan

## Homework Statement

Integrate: (sinx + cosx)/sqrt(1+sin2x)

## Homework Equations

Simple trigo formulae:
cos2x=cos^2(x)--sin^2(x)
sin^2(x)+cos^2(x)=1
sin2x=2.sinx.cosx

## The Attempt at a Solution

I tried to rationalize the given term,multiplying both numerator and denominator with:
1st time:cosx-sinx
2nd: sqrt(1-sin2x)
3rd: Both the above terms
Everytime,I only got slightly different terms.

So,went to substitution.
But,I'm not finding any suitable term which I can substitute.
If after rationalizing,I substitute sin2x as a variable,say,T,things might have worked out,
EXCEPT that since I multiplied the term both to numerator and denominator,the other term just makes it impossible to express the whole term in terms of a simple integral.
Any hints? (With or without substitution)

Do you know about integration by parts and quotient rule integration by parts ?

Last edited:
Tanishq Nandan said:

## Homework Statement

Integrate: (sinx + cosx)/sqrt(1+sin2x)

## Homework Equations

Simple trigo formulae:
cos2x=cos^2(x)--sin^2(x)
sin^2(x)+cos^2(x)=1
sin2x=2.sinx.cosx

## The Attempt at a Solution

I tried to rationalize the given term,multiplying both numerator and denominator with:
1st time:cosx-sinx
2nd: sqrt(1-sin2x)
3rd: Both the above terms
Everytime,I only got slightly different terms.

So,went to substitution.
But,I'm not finding any suitable term which I can substitute.
If after rationalizing,I substitute sin2x as a variable,say,T,things might have worked out,
EXCEPT that since I multiplied the term both to numerator and denominator,the other term just makes it impossible to express the whole term in terms of a simple integral.
Any hints? (With or without substitution)

##(\cos x + \sin x)^2 = ?##

cnh1995
Nidum said:
Do you know about integration by parts and the quotient rule for integration ?
By parts?Yeah,Quotient rule FOR INTEGRATION?I don't think so

Buffu said:
##(\cos x + \sin x)^2 = ?##
1+sin2x,I know,but I already told ya
Tanishq Nandan said:
since I multiplied the term both to numerator and denominator,the other term just makes it impossible to express the whole term in terms of a simple integral.
Then,I have a (sinx + cosx) in the denominator as well,right?That's my problem.
I tried that way as well.

Tanishq Nandan said:

## Homework Statement

Integrate: (sinx + cosx)/sqrt(1+sin2x)

## Homework Equations

Simple trigo formulae:
cos2x=cos^2(x)--sin^2(x)
sin^2(x)+cos^2(x)=1
sin2x=2.sinx.cosx

## The Attempt at a Solution

I tried to rationalize the given term,multiplying both numerator and denominator with:
1st time:cosx-sinx
2nd: sqrt(1-sin2x)
3rd: Both the above terms
If you write 1 as ##\sin^2 x + \cos^2 x##, the denominator becomes
$$\sqrt{ \sin^2 x + \cos^2 x + 2 \sin x \cos x} = \sqrt{(\sin x + \cos x)^2}.$$
Can you see how to simplify ##\sqrt{(\sin x + \cos x)^2}\:?##

Tanishq Nandan
Ray Vickson said:
If you write 1 as ##\sin^2 x + \cos^2 x##, the denominator becomes
$$\sqrt{ \sin^2 x + \cos^2 x + 2 \sin x \cos x} = \sqrt{(\sin x + \cos x)^2}.$$
Can you see how to simplify ##\sqrt{(\sin x + \cos x)^2}\:?##
Ooo...should have thought of that..
K,got it.Thanks!

## 1. What is the basic definition of integration?

Integration is a mathematical process that involves finding the area under a curve or the accumulation of a quantity over a given interval.

## 2. How do you integrate a trigonometric expression like (sinx+cosx)/sqrt(1+sin2x)?

To integrate a trigonometric expression, we use various integration techniques such as substitution, integration by parts, or trigonometric identities. In this case, we can use the substitution method to simplify the expression and then integrate it.

## 3. Can you explain the steps involved in integrating (sinx+cosx)/sqrt(1+sin2x)?

First, we substitute u = 1+sin2x, which simplifies the expression to (sinx+cosx)/sqrt(u). Then, we use the trigonometric identity cos2x = 1-2sin²x to rewrite the expression as 2cosx/sqrt(u). Next, we use the power rule to integrate 1/sqrt(u) and the constant multiple rule to integrate 2cosx. Finally, we substitute back u = 1+sin2x and simplify the expression to get the final answer.

## 4. Is there a shortcut or formula for integrating (sinx+cosx)/sqrt(1+sin2x)?

Yes, there is a formula known as the trigonometric substitution formula that can be used to integrate expressions of the form (sinx+cosx)/sqrt(a²+sin²x). However, in this case, it is easier to use the substitution method as mentioned earlier.

## 5. What are the practical applications of integrating trigonometric expressions like (sinx+cosx)/sqrt(1+sin2x)?

Integrals involving trigonometric functions are used in diverse fields such as physics, engineering, and economics. For example, in physics, integration is used to calculate the displacement, velocity, and acceleration of an object in motion. In engineering, it is used to find the work done, power, and potential energy in a system. In economics, it is used to calculate the marginal cost and revenue of a product.

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