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

Problem

(∫_^)(sin(π*x)*d(x))

Solution

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

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

3. Substitute these into the integral to get (∫_^)(sin(u)1/π*d(u))

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

5. Back-substitute u=π*x to return to the original variable.

Final Answer

(∫_^)(sin(π*x)*d(x))=−cos(π*x)/π+C

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