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Darina [25.2K]
2 years ago
14

Integrate the question below​

Mathematics
1 answer:
Natalka [10]2 years ago
8 0

It looks like the integral is

\displaystyle \int_0^1 \frac{2x - 8x^2}{1+4x} \, dx

Polynomial division yields

\dfrac{2x-8x^2}{1+4x} = -2x + 1 - \dfrac1{1+4x}

Now split up the integral as

\displaystyle \int_0^1 \left(-2x + 1 - \frac1{1+4x}\right) \, dx = \int_0^1 (1-2x) \, dx - \int_0^1 \frac{dx}{1+4x}

In the second integral, substitute u=1+4x and du=4\,dx. Then x=0 \implies u=1 and x=1 \implies u=5, so

\displaystyle \int_0^1 \frac{dx}{1+4x} = \frac14 \int_1^5 \frac{du}u

and by the fundamental theorem of calculus, the integral we want evaluates to

\displaystyle \int_0^1 \left(-2x + 1 - \frac1{1+4x}\right) \, dx = \int_0^1 (1-2x) \, dx - \frac14 \int_1^5 \frac{du}u \\\\ = (x - x^2)\bigg|_{x=0}^{x=1} - \frac14 \ln|u| \bigg|_{u=1}^{u=5} \\\\ = \bigg((1 - 1^2) - (0 - 0^2)\bigg) - \frac14 (\ln(5) - \ln(1)) = \boxed{-\frac{\ln(5)}4}

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Answer:

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Step-by-step explanation:

Using the relation of the rate of each one machine and knowing that adding up both rates.

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Rate S = \frac{x}{18}  'x' is the job

So, using the equation of rates

        Rate total =\frac{x}{36} + \frac{x}{18}

Rate total is going to be a job in 2 hours so = \frac{1}{2}

\frac{1}{2} = \frac{x}{36}+\frac{x}{18} <em>adding up</em>

\frac{1}{2}= x*\frac{3}{36}  <em>factoring the x</em>

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x= \frac{12}{2} \\

x= 6

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avanturin [10]

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Step-by-step explanation:

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Rom4ik [11]

Given:

An athlete who makes 4\dfrac{3}{4} laps in 3 mins 45 seconds on a 400m field.

To find:

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Solution:

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Distance covered in 4\dfrac{3}{4} laps = 4\dfrac{3}{4}\times 400 m

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