HOW WOULD YOU integrate this equation between zero and pi/2
1/(1+(tan^2008)xdx
let u = 2008 x.
Then the integrand becomes
(1/2008)* 1/(1 + tan^2 u) du
= (1/2008)sec^2 u du
The limits of the u integration are 0 to 1004 pi
The integral of sec^x u is tan u
So I get the answer to be (1/2008) tan (1004 pi)
Since 1004 pi is an integer-multiple of 2 pi, the answer is zero.
Check my thinking
I meant to write
The integral of sec^2 u is tan u
The other steps are OK, I believe
To integrate the given equation, 1/(1+tan^2008(x)), between zero and π/2, we will use a substitution method and then apply the limits of integration.
Step 1: Substitution
Let's set u = tan(x). Therefore, du = sec^2(x) dx.
Now, we can rewrite the integral using the substitution u = tan(x) and du = sec^2(x) dx:
∫ 1/(1+tan^2008(x)) dx = ∫ 1/(1+u^2008) * (1/du)
Step 2: Simplification
After substituting, we have:
∫ 1/(1+u^2008) * (1/du)
Step 3: Integration
To integrate this expression, we can use a partial fraction decomposition. However, since the integrand is a higher power, the decomposition becomes very complex and not feasible for manual calculations.
Step 4: Numerical Integration
Considering the complexity of the integrand, it is practical to use numerical integration methods like the Trapezoidal Rule, Simpson's Rule, or numerical software (e.g., MATLAB) to calculate the definite integral between the given limits of integration.
Using numerical integration methods allows us to approximate the integral by dividing the interval into smaller parts and summing up the areas of the corresponding trapezoids or Simpson's rule segments.
Therefore, to find the value of the definite integral between zero and π/2 for 1/(1+tan^2008(x)), we should use numerical integration techniques or software to obtain an approximate solution.