How do you evaluate the integral from 0 to #pi/4# of #(1 + cos^2 x) / (cos^2 x) dx#?
EDIT: The "Funny" response is here!
By signing up, you agree to our Terms of Service and Privacy Policy
To evaluate the integral ( \int_0^{\pi/4} \frac{1 + \cos^2 x}{\cos^2 x} , dx ):
-
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} ).
-
Simplify the expression: ( \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = 1 + \frac{\sin^2 x}{\cos^2 x} ).
-
Notice that ( \frac{\sin^2 x}{\cos^2 x} ) can be rewritten as ( \tan^2 x ).
-
Now the integral becomes: ( \int_0^{\pi/4} \left(1 + \tan^2 x\right) , dx ).
-
Integrate term by term: ( \int_0^{\pi/4} 1 , dx + \int_0^{\pi/4} \tan^2 x , dx ).
-
The integral ( \int_0^{\pi/4} 1 , dx ) evaluates to ( \left[x\right]_0^{\pi/4} = \frac{\pi}{4} - 0 = \frac{\pi}{4} ).
-
To integrate ( \tan^2 x ), use the trigonometric identity ( \tan^2 x = \sec^2 x - 1 ).
-
The integral ( \int \sec^2 x , dx ) is ( \tan x ).
-
So, ( \int_0^{\pi/4} \tan^2 x , dx = \left[\tan x - x\right]_0^{\pi/4} ).
-
Substitute the limits: ( \left[\tan(\pi/4) - \frac{\pi}{4}\right] - \left[\tan(0) - 0\right] ).
-
( \tan(\pi/4) = 1 ) and ( \tan(0) = 0 ), so the result is ( 1 - \frac{\pi}{4} ).
-
Combine the results: ( \frac{\pi}{4} + 1 - \frac{\pi}{4} = 1 ).
Therefore, the value of the given integral is ( 1 ).
By signing up, you agree to our Terms of Service and Privacy Policy
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.
- How do I evaluate the indefinite integral #intcos^5(x)dx# ?
- How do you find the sum of the finite geometric sequence of #Sigma 8(-1/2)^i# from i=0 to 25?
- How do you use limits to evaluate #int x^2dx# from [0,4]?
- How do you find the definite integral of #cot x csc^5 x dx# from #[pi/6, pi/2]#?
- How do you evaluate the integral #int costheta/(1+sintheta)#?
- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7