Evaluate the Integral

Problem

(∫_0^π)(sin(x)*d(x))

Solution

1. Identify the integrand and the limits of integration. The function to integrate is ƒ(x)=sin(x) and the interval is [0,π]

2. Find the antiderivative of sin(x) Recall that the derivative of cos(x) is −sin(x) so the antiderivative of sin(x) is −cos(x)

3. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit π and subtracting the value at the lower limit 0

[−cos(x)]π0

4. Substitute the values into the expression.

−cos(π)−(−cos(0))

5. Simplify the trigonometric values. Since cos(π)=−1 and cos(0)=1

−(−1)−(−1)

1+1=2

Final Answer

(∫_0^π)(sin(x)*d(x))=2

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