Evaluate the Integral

Problem

(∫_π/3^π)(sin(x)*d(x))

Solution

1. Identify the integral as a definite integral of the sine function over the interval [π/3,π]

2. Find the antiderivative of sin(x) which is −cos(x)

3. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit π and the lower limit π/3

[−cos(x)]π(π/3)

4. Substitute the limits into the expression.

(−cos(π))−(−cos(π/3))

5. Evaluate the trigonometric values where cos(π)=−1 and cos(π/3)=1/2

−(−1)−(−1/2)

6. Simplify the resulting numerical expression.

1+1/2=3/2

Final Answer

(∫_π/3^π)(sin(x)*d(x))=3/2

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