Use the fundamental theorem of calculus to evaluate the following integral
∫^π/2 **the bottom is zeroe. |cosx| dx
To evaluate the integral ∫₀^(π/2) |cos(x)| dx using the fundamental theorem of calculus, we can follow these steps:
Step 1: Identify the antiderivative of the integrand.
In this case, the function inside the absolute value, |cos(x)|, is equal to cos(x) for x ∈ [0, π/2]. Therefore, we don't need to consider the absolute value function, and its antiderivative is sin(x).
Step 2: Evaluate the antiderivative at the upper limit and lower limit.
Using the antiderivative we found in Step 1 (sin(x)), evaluate it at the upper limit of integration (π/2) and subtract the evaluated antiderivative at the lower limit of integration (0) to get the final result.
∫₀^(π/2) |cos(x)| dx = sin(π/2) - sin(0)
Step 3: Simplify the expression.
sin(π/2) evaluates to 1, and sin(0) evaluates to 0.
∫₀^(π/2) |cos(x)| dx = 1 - 0
Step 4: Simplify further if necessary.
The final result is:
∫₀^(π/2) |cos(x)| dx = 1
Therefore, the value of the integral ∫₀^(π/2) |cos(x)| dx is 1.