Evaluate the Integral

Problem

(∫_^)(sin(4*x)*cos(4*x)*d(x))

Solution

1. Identify a trigonometric identity to simplify the integrand. The double-angle identity for sine is sin(2*θ)=2*sin(θ)*cos(θ)

2. Rearrange the identity to solve for the product: sin(θ)*cos(θ)=1/2*sin(2*θ)

3. Substitute θ=4*x into the identity, which gives sin(4*x)*cos(4*x)=1/2*sin(8*x)

4. Rewrite the integral using this substitution.

(∫_^)(1/2*sin(8*x)*d(x))

5. Apply the constant multiple rule by moving the 1/2 outside the integral.

1/2*(∫_^)(sin(8*x)*d(x))

6. Integrate using the rule (∫_^)(sin(a*x)*d(x))=−1/a*cos(a*x)+C where a=8

1/2*(−1/8*cos(8*x))+C

7. Simplify the expression by multiplying the fractions.

−1/16*cos(8*x)+C

Final Answer

(∫_^)(sin(4*x)*cos(4*x)*d(x))=−1/16*cos(8*x)+C

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