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vladimir2022 [97]

2 years ago

7

Oliver is driving 360 miles,which is 136 miles less than twice the distance Matt is driving. write an equation that can be used

to determine the distance in miles Matt is driving.

1 answer:

Minchanka [31]2 years ago

7 0

The answer would be (2x-136)=360

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ivann1987 [24]

Answer: 1 Brutus Bites= $9.25

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2 years ago

∫<img src="https://tex.z-dn.net/?f=%5Cfrac%7Bcosxdx%7D%7B%5Csqrt%7B1%2Bsin%5E%7B2%7Dx%20%7D%20%7D" id="TexFormula1" title="\frac

const2013 [10]

Substitute sin(x) = tan(t) and cos(x) dx = sec²(t) dt. We want this change of variable to be reversible, so let's assume bot x and t are bounded between 0 and π/2.

Then we have

$\displaystyle \int \frac{\cos(x)}{\sqrt{1 + \sin^2(x)}} \, dx = \int \frac{\sec^2(t)}{\sqrt{1 + \tan^2(t)}} \, dt$

Recall the Pythagorean identity,

1 + tan²(t) = sec²(t)

Then

√(1 + tan²(t)) = √(sec²(t)) = sec(t)

and the integral reduces to

$\displaystyle \int \frac{\sec^2(t)}{\sqrt{1 + \tan^2(t)}} \, dt = \int \frac{\sec^2(t)}{\sec(t)} \, dt = \int \sec(t) \, dt = \ln|\sec(t)+\tan(t)| + C$

Change the variable back to x, so the antiderivative is

$\displaystyle \int \frac{\cos(x)}{\sqrt{1 + \sin^2(x)}} \, dx = \ln \left|\sec\left(\tan^{-1}(\sin(x))\right) + \tan\left(\tan^{-1}(\sin(x))\right) \right| + C$

$\displaystyle \int \frac{\cos(x)}{\sqrt{1 + \sin^2(x)}} \, dx = \boxed{\ln \left|\sqrt{1+\sin^2(x)} + \sin(x) \right| + C}$

6 0

2 years ago