# How do you evaluate the definite integral #int cosxe^sinx# from #[0, pi/2]#?

recognising that # d/(dx)(e^sinx)=cosxe^sinx

we have

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To evaluate the definite integral (\int_{0}^{\frac{\pi}{2}} \cos(x)e^{\sin(x)} ,dx), you can use the technique of integration by parts. Let (u = \cos(x)) and (dv = e^{\sin(x)},dx). Then, (du = -\sin(x),dx) and (v = -e^{\sin(x)}).

Now, applying the integration by parts formula, (\int u ,dv = uv - \int v ,du), we get:

[
\begin{align*}
\int_{0}^{\frac{\pi}{2}} \cos(x)e^{\sin(x)} ,dx &= \left. -\cos(x)e^{\sin(x)} \right|*{0}^{\frac{\pi}{2}} - \int*{0}^{\frac{\pi}{2}} (-e^{\sin(x)})\sin(x) ,dx \
&= -\cos\left(\frac{\pi}{2}\right)e^{\sin\left(\frac{\pi}{2}\right)} + \cos(0)e^{\sin(0)} - \int_{0}^{\frac{\pi}{2}} (-e^{\sin(x)})\sin(x) ,dx \
&= 0 - e^0 - \int_{0}^{\frac{\pi}{2}} (-e^{\sin(x)})\sin(x) ,dx \
&= -1 - \int_{0}^{\frac{\pi}{2}} (-e^{\sin(x)})\sin(x) ,dx
\end{align*}
]

Now, for the remaining integral, (\int_{0}^{\frac{\pi}{2}} (-e^{\sin(x)})\sin(x) ,dx), there's no straightforward method to evaluate it analytically. Thus, we might resort to numerical methods like Simpson's rule or the trapezoidal rule to approximate its value.

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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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