Evaluate the Integral integral of sin(3x) with respect to x

Problem

(∫_^)(sin(3*x)*d(x))

Solution

1. Identify the integral as a basic trigonometric integral requiring a usubstitution or the reverse chain rule for a linear argument.

2. Apply the substitution u=3*x which implies that d(u)=3*d(x) or d(x)=1/3*d(u)

3. Substitute these values into the integral to change the variable from x to u

(∫_^)(sin(u)⋅1/3*d(u))

4. Factor out the constant 1/3 from the integral.

1/3*(∫_^)(sin(u)*d(u))

5. Integrate the sine function, noting that the integral of sin(u) is −cos(u)

−1/3*cos(u)+C

6. Back-substitute u=3*x to express the final result in terms of x

−1/3*cos(3*x)+C

Final Answer

(∫_^)(sin(3*x)*d(x))=−1/3*cos(3*x)+C

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