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How do you evaluate the definite integral #int e^cosx*sinx# from #[0,pi/4]#?

Answer 1

Isaiah Allen

# = e - e^(1/sqrt 2)#

#int_0^(pi/4) e^cosx*sinx \ dx#

#= int_0^(pi/4) d/dx (color(red)(-) e^cosx) \ dx#

# = [- e^cosx]_0^(pi/4) #

# = e - e^(1/sqrt 2)#

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Answer 2

Emily Ammerman

To evaluate the definite integral (\int_{0}^{\frac{\pi}{4}} e^{\cos(x)} \sin(x) , dx), you can use substitution. Let (u = \cos(x)), then (du = -\sin(x) , dx). The limits of integration also change accordingly:

When (x = 0), (u = \cos(0) = 1).
When (x = \frac{\pi}{4}), (u = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}).

So, the integral becomes:

(\int_{0}^{\frac{\pi}{4}} e^{\cos(x)} \sin(x) , dx = -\int_{1}^{\frac{\sqrt{2}}{2}} e^u , du)

Now, integrate (e^u) with respect to (u) and evaluate it at the limits:

(-\int_{1}^{\frac{\sqrt{2}}{2}} e^u , du = -e^u \bigg|_{1}^{\frac{\sqrt{2}}{2}} = -\left(e^{\frac{\sqrt{2}}{2}} - e^1\right))

Therefore, the value of the definite integral is (-\left(e^{\frac{\sqrt{2}}{2}} - e\right)).

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Answer from HIX Tutor

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.