# How do you find the definite integral for: #10sin(x)dx# for the intervals #[0, pi]#?

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To find the definite integral of (10 \sin(x)) with respect to (x) over the interval ([0, \pi]), you can use the definite integral formula:

[ \int_{a}^{b} f(x) , dx ]

where (a) and (b) are the limits of integration. In this case, (a = 0) and (b = \pi).

The integral of (10 \sin(x)) with respect to (x) is:

[ \int 10 \sin(x) , dx = -10 \cos(x) + C ]

where (C) is the constant of integration.

To find the definite integral over the interval ([0, \pi]), you evaluate the antiderivative at the upper limit and subtract the antiderivative evaluated at the lower limit:

[ \int_{0}^{\pi} 10 \sin(x) , dx = \left[-10 \cos(x)\right]_{0}^{\pi} ]

[ = -10 \cos(\pi) - (-10 \cos(0)) ]

[ = -10(-1) - (-10(1)) ]

[ = 10 - (-10) ]

[ = 10 + 10 ]

[ = 20 ]

So, the definite integral of (10 \sin(x)) with respect to (x) over the interval ([0, \pi]) is (20).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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