The integral   \int_{\frac{\pi }{4}}^{\frac{3\pi }{4}}\frac{dx}{1+\cos x} 

is equal to :

 

  • Option 1)

    2

  • Option 2)

    4

  • Option 3)

    -1

  • Option 4)

    -2

 

Answers (1)

As learnt in concept

Properties of Definite integration -

\int_{a}^{b}f\left ( x \right )dx= \int_{a}^{b}f\left ( a+b-x \right )dx

When \int_{0}^{b}f\left ( x \right )dx= \int_{0}^{b}f\left ( b-x \right )dx

 

- wherein

Put the \left ( a+b-x \right ) at the place of x in f\left ( x \right )

 

 I=\int_{\frac{\pi }{4}}^{\frac{3\pi }{4}}\frac{dx}{1+\cos x}

Also, I=\int_{\frac{\pi }{4}}^{\frac{3\pi }{4}} \frac{dx}{1+\cos\left ( \pi -x \right ) }

=\int_{\frac{\pi }{4}}^{\frac{3\pi }{4}}\frac{dx}{1-\cos x}

2I=\int_{\frac{\pi }{4}}^{\frac{3\pi }{4}}dx\left ( \frac{1}{1+\cos x}+\frac{1}{1-\cos x} \right )

2I=\int_{\frac{\pi }{4}}^{\frac{3\pi }{4}}\frac{2}{\sin ^{2}x} dx

I=\int_{\frac{\pi }{4}}^{\frac{3\pi }{4}} cosec^{2}x\: dx

=\left [-\cot x \right ]^{\frac{3\pi }{4}}_{\frac{\pi }{4}}

=-\left( ( -1 \right )-\left (1 \right ))

= 2


Option 1)

2

Correct option

Option 2)

4

Incorrect option    

Option 3)

-1

Incorrect option    

Option 4)

-2

Incorrect option    

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