Final answer:
The antiderivative of sin(πx) is –(cos(πx))/π + C, using the basic integration rule ∫ sin(ax) dx = –cos(ax)/a + C.
Step-by-step explanation:
To find the antiderivative of sin(πx), we need to apply integral calculus techniques. The solution involves recognizing that the function is a sinusoidal function and using the fundamental rules of integration.
First, let us consider the integral of sin(πx). The antiderivative of sin(ax), where a is a constant, is –(cos(ax))/a + C, where C is the constant of integration. So, the antiderivative of sin(πx) will follow the same pattern:
- ∫ sin(πx) dx = –(cos(πx))/π + C
This result uses the basic integration rule that ∫ sin(ax) dx = –cos(ax)/a + C. It's also useful to remember that when integrating trigonometric functions, it's often required to include a coefficient related to the inner function's derivative, in this case, π, to compensate for the chain rule during differentiation.
As an example, if we needed to calculate the definite integral of sin(πx) between limits that correspond to one period of the function, we would find that the integral evaluates to zero since sin(πx) is an odd function about the y-axis and its integral over a symmetric interval around the origin yields zero due to the areas above and below the x-axis cancelling each other out.