Evaluate the integral upper limit 7pi/4 and lower limit 5pi/4 of 10 csc theta cot theta dtheta
To evaluate the integral ∫(5π/4 to 7π/4) 10 csc(θ) cot(θ) dθ, we can use the trigonometric identity:
csc(θ) = 1/sin(θ) and cot(θ) = 1/tan(θ) = cos(θ)/sin(θ).
Let's rewrite the integral using these identities:
∫(5π/4 to 7π/4) 10 csc(θ) cot(θ) dθ
= ∫(5π/4 to 7π/4) 10 (1/sin(θ)) (cos(θ)/sin(θ)) dθ.
Now, let's simplify the expression further:
∫(5π/4 to 7π/4) 10 (cos(θ))/(sin^2(θ)) dθ.
Next, we can make a substitution to simplify the integral. Let's substitute u = sin(θ), so du = cos(θ) dθ.
When θ = 5π/4, sin(θ) = sin(5π/4) = -√2/2.
When θ = 7π/4, sin(θ) = sin(7π/4) = -√2/2.
So, the new limits of integration in terms of u become:
u = sin(5π/4) = -√2/2 to u = sin(7π/4) = -√2/2.
The integral becomes:
∫(-√2/2 to -√2/2) 10 du/ u^2
Since the limits of integration are the same, the integral evaluates to zero:
∫(-√2/2 to -√2/2) 10 du/ u^2 = 0.
Therefore, the value of the given integral from 5π/4 to 7π/4 is 0.