How do you evaluate the definite integral #int 1+sinx# from #[pi/4, pi/2]#?
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To evaluate the definite integral (\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (1 + \sin(x)) , dx), you can proceed as follows:

Integrate (1) with respect to (x) within the given limits: [\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} 1 , dx = x \Bigg_{\frac{\pi}{4}}^{\frac{\pi}{2}} = \frac{\pi}{2}  \frac{\pi}{4} = \frac{\pi}{4}]

Integrate (\sin(x)) with respect to (x) within the given limits: [\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \sin(x) , dx = \cos(x) \Bigg_{\frac{\pi}{4}}^{\frac{\pi}{2}} = \cos\left(\frac{\pi}{2}\right) + \cos\left(\frac{\pi}{4}\right) = 0 + \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}]

Add the results from step 1 and step 2: [\frac{\pi}{4} + \frac{\sqrt{2}}{2}]
So, [\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (1 + \sin(x)) , dx = \frac{\pi}{4} + \frac{\sqrt{2}}{2}]
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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