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\int \frac{\cos x-\sin x}{\cos x+\sin x}\left ( 2+2\sin x \right )dx is equal to

  • Option 1)

    sin2x+c

  • Option 2)

    cos2x+c

  • Option 3)

    tan2x+c

  • Option 4)

    none

 

Answers (1)

best_answer

As we learnt

Type of Integration by perfect square -

Integrals in the form of     \int \frac{p\cos x+q\sin x }{a\cos x+b\sin x}dx

- wherein

Working rule :

p\cos x+q\sin x=A\cdot \frac{\mathrm{d} }{\mathrm{d} x}(a\cos x+b\sin x)+B(a\cos x+b\sin x) 

Find A and B by comparing sinx and cosx 

 

  \int {\frac{{\cos x - \sin x}}{{\cos x + \sin x}}} .\left( {2 + 2\sin 2x} \right)\,dx = 2\int {\frac{{\left( {\cos x - \sin x} \right)\,{{\left( {\cos x + \sin x} \right)}^2}}}{{\cos x + \sin x}}} \,dx

=2\int {\left( {\cos x - \sin x} \right)\,\left( {\cos x + \sin x} \right)} \,dx$

=2\int {\left( {{{\cos }^2}x - {{\sin }^2}x} \right)\,dx = } 2\int {\cos 2x} \,dx=\sin 2x + c

 


Option 1)

sin2x+c

Option 2)

cos2x+c

Option 3)

tan2x+c

Option 4)

none

Posted by

Aadil

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