How do you evaluate the integral from 0 to #pi/4# of #(1 + cos^2 x) / (cos^2 x) dx#?

Answer 1
Just do simple math : #int_0^(pi/4)(1+cos^2(x))/cos^2(x) dx #
#=int_0^(pi/4)1/cos^2(x)+1 dx#
#=(tan(x)+x)~|_0^(pi/4)#
#=1+pi/4#

EDIT: The "Funny" response is here!

#int_0^(pi/4)(1+cos^2(x))/cos^2(x)dx = int_0^(pi/4)(2+cos^2(x)-1)/cos^2(x)dx#
Factorize : #int_0^(pi/4)(-(1-cos^2(x))+2)/cos^2(x)dx = int_0^(pi/4)(-sin^2(x)+2)/cos^2(x)dx#
#=-int_0^(pi/4)sin^2(x)/cos^2(x) dx+2int_0^(pi/4)1/cos^2(x) dx#
#=-int_0^(pi/4)1+tan^2(x)-1 dx + 2int_0^(pi/4)1/cos^2(x)dx#
#=-(tan(x)-x)~|_0^(pi/4)+2(tan(x))~|_0^(pi/4)#
#=-1+pi/4+2 = 1+pi/4#
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Answer 2

To evaluate the integral ( \int_0^{\pi/4} \frac{1 + \cos^2 x}{\cos^2 x} , dx ):

  1. Rewrite the integrand using a trigonometric identity: ( \frac{1 + \cos^2 x}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} ).

  2. Simplify the expression: ( \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = 1 + \frac{\sin^2 x}{\cos^2 x} ).

  3. Notice that ( \frac{\sin^2 x}{\cos^2 x} ) can be rewritten as ( \tan^2 x ).

  4. Now the integral becomes: ( \int_0^{\pi/4} \left(1 + \tan^2 x\right) , dx ).

  5. Integrate term by term: ( \int_0^{\pi/4} 1 , dx + \int_0^{\pi/4} \tan^2 x , dx ).

  6. The integral ( \int_0^{\pi/4} 1 , dx ) evaluates to ( \left[x\right]_0^{\pi/4} = \frac{\pi}{4} - 0 = \frac{\pi}{4} ).

  7. To integrate ( \tan^2 x ), use the trigonometric identity ( \tan^2 x = \sec^2 x - 1 ).

  8. The integral ( \int \sec^2 x , dx ) is ( \tan x ).

  9. So, ( \int_0^{\pi/4} \tan^2 x , dx = \left[\tan x - x\right]_0^{\pi/4} ).

  10. Substitute the limits: ( \left[\tan(\pi/4) - \frac{\pi}{4}\right] - \left[\tan(0) - 0\right] ).

  11. ( \tan(\pi/4) = 1 ) and ( \tan(0) = 0 ), so the result is ( 1 - \frac{\pi}{4} ).

  12. Combine the results: ( \frac{\pi}{4} + 1 - \frac{\pi}{4} = 1 ).

Therefore, the value of the given integral is ( 1 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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