Evaluate the Integral

Problem

(∫_0^π)(cos2(x)*d(x))

Solution

1. Apply the power-reduction identity to rewrite the integrand using the formula cos2(x)=(1+cos(2*x))/2

(∫_0^π)((1+cos(2*x))/2*d(x))

2. Factor out the constant 1/2 from the integral to simplify the expression.

1/2*(∫_0^π)((1+cos(2*x))*d(x))

3. Find the antiderivative of each term in the integrand.

1/2*[x+sin(2*x)/2]π0

4. Evaluate the definite integral by substituting the upper limit π and the lower limit 0 into the antiderivative.

1/2*((π+sin(2*π)/2)−(0+sin(0)/2))

5. Simplify the trigonometric values noting that sin(2*π)=0 and sin(0)=0

1/2*(π+0−0−0)

6. Calculate the final result by multiplying the remaining terms.

π/2

Final Answer

(∫_0^π)(cos2(x)*d(x))=π/2

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