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

[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)

 

[Nidum]

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

 

[Buffu]

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)

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

 

[Tanishq Nandan]

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

 

[Tanishq Nandan]

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

1+sin2x,I know,but I already told ya

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.

 

[Ray Vickson]

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]

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!