Evaluate the Integral integral of sin(8x) with respect to x

Problem

(∫_^)(sin(8*x)*d(x))

Solution

1. Identify the integral form as a basic trigonometric integral requiring a u-substitution for the linear argument 8*x

2. Substitute u=8*x which implies that the derivative is d(u)/d(x)=8 or d(x)=1/8*d(u)

3. Rewrite the integral in terms of u by substituting the expressions for 8*x and d(x)

(∫_^)(sin(u)⋅1/8*d(u))

4. Factor out the constant 1/8 from the integral.

1/8*(∫_^)(sin(u)*d(u))

5. Integrate the sine function using the rule (∫_^)(sin(u)*d(u))=−cos(u)+C

1/8*(−cos(u))+C

6. Back-substitute the original expression 8*x for u to get the final result.

−1/8*cos(8*x)+C

Final Answer

(∫_^)(sin(8*x)*d(x))=−1/8*cos(8*x)+C

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