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adoni [48]
3 years ago
14

Integrate the following

Mathematics
1 answer:
enot [183]3 years ago
7 0

I suppose you mean to have the entire numerator under the square root?

\displaystyle\int_2^4\frac{\sqrt{x^2-4}}{x^2}\,\mathrm dx

We can use a trigonometric substitution to start:

x=2\sec t\implies\mathrm dx=2\sec t\tan t\,\mathrm dt

Then for x=2, t=\sec^{-1}1=0; for x=4, t=\sec^{-1}2=\frac\pi3. So the integral is equivalent to

\displaystyle\int_0^{\pi/3}\frac{\sqrt{(2\sec t)^2-4}}{(2\sec t)^2}(2\sec t\tan t)\,\mathrm dt=\int_0^{\pi/3}\frac{\tan^2t}{\sec t}\,\mathrm dt

We can write

\dfrac{\tan^2t}{\sec t}=\dfrac{\frac{\sin^2t}{\cos^2t}}{\frac1{\cos t}}=\dfrac{\sin^2t}{\cos t}=\dfrac{1-\cos^2t}{\cos t}=\sec t-\cos t

so the integral becomes

\displaystyle\int_0^{\pi/3}(\sec t-\cos t)\,\mathrm dt=\boxed{\ln(2+\sqrt3)-\frac{\sqrt3}2}

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Answer:

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Step-by-step explanation:

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5 0
2 years ago
Vertex of an absolute value graph ______ of the graph.
Inga [223]

Answer:

is the axis of symmetry

Step-by-step explanation:

i believe this is the answer :)

5 0
3 years ago
A car's speedometer has a percent error of 5%. The speedometer currently shows that the car is traveling at a speed of 60 mph.
Minchanka [31]

Answer:

The actual speed of the car is either 57 mph or 63 mph

Step-by-step explanation:

Given as :

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Let The actual speed of car = x mph

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So, \frac{5}{100} = \frac{x - 60}{60}

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Or, 3 = x - 60

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8 0
3 years ago
I bought 16 cupcakes and then last week i bought 9 keep the pattern going
lana [24]

Answer:

25

Step-by-step explanation:

4 0
2 years ago
A text message plan costs ​$4 per month plus ​$0.32 per text. Find the monthly cost for x text messages.
krok68 [10]

Answer:

4 + 0.32x  

Step-by-step explanation:

       0.32 = cost of one text message

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6 0
3 years ago
Read 2 more answers
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