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  • How to solve this problem- - Integral Calculus - JEE Main

The value of the integral     \int_{\frac{-\pi }{2}}^{\frac{\pi }{2}}\sin ^{4}x\left [ 1+\log \left [ \frac{2+\sin x}{2-\sin x} \right ] \right ]dx   is :

 

  • Option 1)

    0

     

     

     

  • Option 2)

    \frac{3}{4}

  • Option 3)

    \frac{3}{8}\pi

  • Option 4)

    \frac{3}{16}\pi

 

Answers (2)

best_answer

As we learned

 

Properties of definite integration -

If f\left ( x \right ) is an EVEN function of x: then integral of the function from - a to a is the same as twice the integral of the same function from o to a.

\int_{-a}^{a}f(x)dx= 2\left \{ \int_{o}^{a} f(x)dx\right \}

 

- wherein

Check even function f(-x)=f(x) and symmetrical about y axis.

 

 

 

Properties of Definite Integration -

If f(x) is an odd function of x then integral of the function from -a to a is ZERO

\int_{-a}^{a}f(x)dx=0
 

- wherein

Check

Odd function f(-x)= -f(x)

 

 

\int_{\frac{-\pi }{2}}^{\frac{\pi }{2}}\sin ^{4}x\left ( 1+\log \left ( \frac{2+\sin x}{2-\sin x} \right ) \right )dx

=2\int_{0}^{\frac{\pi }{2}}\sin ^{4}xdx+2\int_{0}^{\frac{\pi }{2}}

=2\times \frac{3.1}{4.2}\times \frac{\pi }{2}=3\frac{\pi }{8}

 

 


Option 1)

0

 

 

 

Option 2)

\frac{3}{4}

Option 3)

\frac{3}{8}\pi

Option 4)

\frac{3}{16}\pi

Posted by

Himanshu

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