Evaluate the following definite integral:∫𝜋30sin(𝑥) d𝑥=

Sure, here are the steps to evaluate the definite integral ∫𝜋/3 0 sin(𝑥) d𝑥:

Step 1: Identify the antiderivative of sin(x), which is -cos(x).

Step 2: Apply the Fundamental Theorem of Calculus, which states that the definite integral of a function from a to b is equal to the antiderivative evaluated at b minus the antiderivative evaluated at a.

So, ∫𝜋/3 0 sin(𝑥) d𝑥 = [-cos(x)] from 0 to 𝜋/3.

Step 3: Substitute the limits of integration into the antiderivative:

= -cos(𝜋/3) - (-cos(0))

Step 4: Evaluate the cosine values:

= -1/2 - (-1)

Step 5: Simplify the expression:

= -1/2 + 1 = 1/2

So, ∫𝜋/3 0 sin(𝑥) d𝑥 = 1/2.