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Nataly_w [17]
1 year ago
11

Integration by Parts Evaluate e-2x cos(2x) dx.​

Mathematics
1 answer:
kifflom [539]1 year ago
6 0

Let

I = \displaystyle \int e^{-2x} \cos(2x) \, dx[/]texIntegrate by parts:[tex]\displaystyle \int u \, dv = uv - \int v \, du

with

u = e^{-2x} \implies du = -2 e^{-2x} \, dx \\\\ dv = \cos(2x) \, dx \implies v = \dfrac12 \sin(2x)

Then

\displaystyle I = \frac12 e^{-2x} \sin(2x) + \int e^{-2x} \sin(2x) \, dx + C

Integrate by parts again, this time with

u = e^{-2x} \implies du = -2 e^{-2x} \, dx \\\\ dv = \sin(2x) \, dx \implies v = -\dfrac12 \cos(2x)

so that

\displaystyle I = \frac12 e^{-2x} \sin(2x) - \frac12 e^{-2x} \cos(2x) - \int e^{-2x} \cos(2x) \, dx + C\\\\ \implies I = \frac{\sin(2x)-\cos(2x)}{2e^{2x}} - I + C \\\\ \implies 2I = \frac{\sin(2x) - \cos(2x)}{2e^{2x}} + C \\\\ \implies I = \boxed{\frac{\sin(2x) - \cos(2x)}{4e^{2x}} + C}

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Yanka [14]

Answer:

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the range of the function f(x) is f(x)\ge 9

Step-by-step explanation:

Consider the parent function y=\sqrt{x}.

The domain og this function is x\ge 0, the range of this function is y\ge 0.

The function f(x)=\sqrt{x-7}+9 is translated function y=\sqrt{x} 7 units to the right and 9 units up, so

the domain of the function f(x) is x\ge 7

the range of the function f(x) is f(x)\ge 9

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3 years ago
What is the solution to this inequality?<br>4(2x + 2) + 12 &gt; 100​
Umnica [9.8K]

Answer:

Step-by-step explanation:

x>10

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3 years ago
In the figure, angle ZYX is measured in degrees. The area of the shaded sector can be determined using the formula StartFraction
tensa zangetsu [6.8K]

<u>The given options are:</u>

(A)the central angle measure of the sector divided by the total angle measure of a circle multiplied by the area of the circle will yield the area of the sector.

(B)the central angle measure of the sector divided by the total angle measure of a circle multiplied by the circumference of the circle will yield the area of the sector.

(C)the central angle measure of the sector multiplied by the area of the circle will yield the area of the sector.

(D)the central angle measure of the sector multiplied by the circumference of the circle will yield the area of the sector.

Answer:

(A)the central angle measure of the sector divided by the total angle measure of a circle multiplied by the area of the circle will yield the area of the sector.

Step-by-step explanation:

The area of the shaded sector can be determined using the formula:

\frac{m\angle ZYX}{360^\circ} \cdot \pi r^2

m\angle XYZ=$Central angle of XYZ\\\pi r^2$=Area of a Circle

360^\circ =$Total Angle

Therefore, the formula is:

\dfrac{m\angle ZYX}{360^\circ} \cdot \pi r^2\\\text{Area of XYZ}=\dfrac{\text{Cental Angle of XYZ}}{\text{Total Angle}} X \text{Area of the Circle}

Therefore, the formula is best explained by Option A.

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Which statement is true about the function f(x)=-√-x
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Answer: D.The domain and range of the function are the same.


Step-by-step explanation: Given function f(x)=-√-x.

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In order to get a positive number or 0 inside square root, let us solve the inequality:

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On dividing both sides by -1, the inequality sign would get flip and we get

<h3>Domain: x \leq 0.</h3>

We can see that we have a negative sign in front of square root. So, the value of f(x) would be always a negative number or 0.

<h3>Therefore, range would also be :x \leq 0..</h3><h3>Therefore, correct option is D option.</h3>
5 0
3 years ago
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