# How do you evaluate the definite integral #int e^(sin(x))*cosx dx# from #[0,pi]#?

We have

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To evaluate the definite integral ( \int_{0}^{\pi} e^{\sin(x)} \cdot \cos(x) , dx ) from (0) to (\pi), we can use the substitution method.

Let ( u = \sin(x) ). Then, ( du = \cos(x) , dx ).

When ( x = 0 ), ( u = \sin(0) = 0 ), and when ( x = \pi ), ( u = \sin(\pi) = 0 ).

So, the integral becomes:

[ \int_{0}^{\pi} e^{\sin(x)} \cdot \cos(x) , dx = \int_{0}^{0} e^u , du ]

Since the limits of integration become the same after the substitution, the integral becomes (0).

Therefore, the value of the definite integral is (0).

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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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