Understanding and Evaluating the Integral of Sin(x) with Respect to x and the Importance of the Constant of Integration

∫sinxdx

To evaluate the integral of sin(x) with respect to x, we can use integration techniques

To evaluate the integral of sin(x) with respect to x, we can use integration techniques. The integral of sin(x) is given by:

∫sin(x) dx = -cos(x) + C

Where C is the constant of integration.

To understand how we obtained this result, let’s briefly discuss the process.

Integration is the reverse operation of differentiation. We apply integration rules to find the antiderivative of a function. In this case, we recognize that the derivative of -cos(x) with respect to x is sin(x). Therefore, -cos(x) is the antiderivative of sin(x).

The “+ C” in the result represents the constant of integration since the derivative of any constant is zero. It is required because when we take the derivative of -cos(x), the constant term disappears. The “C” allows us to account for all possible functions whose derivative is sin(x).

So, the final answer to the integral of sin(x) is:
∫sin(x) dx = -cos(x) + C

In summary, the integral of sin(x) with respect to x is -cos(x) plus a constant C.

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