Evaluate the Integral integral of sin(4x) with respect to x

Problem

(∫_^)(sin(4*x)*d(x))

Solution

1. Identify the integral as a basic trigonometric integral requiring a usubstitution because the argument of the sine function is 4*x rather than just x

2. Substitute u=4*x to simplify the integrand.

3. Differentiate the substitution to find the relationship between d(u) and d(x)

d(u)/d(x)=4

d(x)=1/4*d(u)

4. Rewrite the integral in terms of u

(∫_^)(sin(u)⋅1/4*d(u))

1/4*(∫_^)(sin(u)*d(u))

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

1/4*(−cos(u))+C

6. Back-substitute 4*x for u to return to the original variable:

−1/4*cos(4*x)+C

Final Answer

(∫_^)(sin(4*x)*d(x))=−1/4*cos(4*x)+C

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