Evaluate the Integral

Problem

(∫_^)((1−cos(2*x))/2*d(x))

Solution

1. Split the integral into two separate terms by distributing the denominator.

(∫_^)(1/2*d(x))−(∫_^)(cos(2*x)/2*d(x))

2. Factor out the constant coefficients from each integral.

1/2*(∫_^)(1*d(x))−1/2*(∫_^)(cos(2*x)*d(x))

3. Integrate the first term using the power rule for constants.

1/2*(∫_^)(1*d(x))=1/2*x

4. Integrate the second term using the substitution u=2*x which results in a factor of 1/2

1/2*(∫_^)(cos(2*x)*d(x))=1/2⋅sin(2*x)/2

5. Combine the results and add the constant of integration C

1/2*x−1/4*sin(2*x)+C

Final Answer

(∫_^)((1−cos(2*x))/2*d(x))=x/2−sin(2*x)/4+C

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