Evaluate the Integral

Problem

(∫_0^π/2)(sin(2*x)*d(x))

Solution

1. Identify the integral as a definite integral of a trigonometric function requiring the reverse chain rule or usubstitution.

2. Find the antiderivative of sin(2*x) by using the rule (∫_^)(sin(a*x)*d(x))=−1/a*cos(a*x)

3. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit π/2 and the lower limit 0

F(x)=−1/2*cos(2*x)

4. Substitute the upper limit into the antiderivative.

−1/2*cos(2⋅π/2)=−1/2*cos(π)

5. Substitute the lower limit into the antiderivative.

−1/2*cos(2⋅0)=−1/2*cos(0)

6. Calculate the difference between the upper and lower limit values.

(−1/2⋅−1)−(−1/2⋅1)

7. Simplify the resulting numerical expression.

1/2+1/2=1

Final Answer

(∫_0^π/2)(sin(2*x)*d(x))=1

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