Evaluate the Integral integral of 2sin(x)^2 with respect to x

Problem

(∫_^)(2*sin2(x)*d(x))

Solution

1. Identify the trigonometric identity for sin2(x) to simplify the integrand.

2. Apply the identity sin2(x)=(1−cos(2*x))/2 to rewrite the expression.

2*sin2(x)=2⋅(1−cos(2*x))/2

3. Simplify the expression by canceling the constant factor of 2

2*sin2(x)=1−cos(2*x)

4. Substitute the simplified expression back into the integral.

(∫_^)((1−cos(2*x))*d(x))

5. Integrate each term separately using the power rule and the basic integral of cosine.

(∫_^)(1*d(x))−(∫_^)(cos(2*x)*d(x))

6. Evaluate the integrals, remembering to divide by the coefficient of x for the trigonometric term and add the constant of integration C

x−sin(2*x)/2+C

Final Answer

(∫_^)(2*sin2(x)*d(x))=x−sin(2*x)/2+C

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