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DiKsa [7]
3 years ago
5

The standard form of the number 5, 976,000,000,000,000,000,000,000 is​

Mathematics
1 answer:
nirvana33 [79]3 years ago
5 0

Answer:

5.976\times10^{24}

Step-by-step explanation:

<em>Standard form</em>, also called <em>scientific notation</em>, is a more compact way of writing very big or very small numbers. All numbers in scientific notation have the form

m\times10^b

where m is a number between 1 and 10, and b is an integer that represents how we've shifted the decimal point from the original number. For example, the number 23,000 in standard form would be written as 2.3 × 10⁴, since we have to shift the decimal point 4 places to the right to get back our original number.

We can turn 5,976,000,000,000,000,000,000,000 into a number between 1 and 10 by shifting the decimal 24 places to the left, giving us 5.976. We'll save that 24 as the exponent for our 10, giving us the final form

5.976\times10^{24}

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Compute the definite integral:
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Evaluate the antiderivative at the limits and subtract.
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For the integrand 1/(x^2 + 3 x + 2), complete the square:
 = (5 log(3))/2 + 1/2 integral_0^1 1/((x + 3/2)^2 - 1/4) dx

For the integrand 1/((x + 3/2)^2 - 1/4), substitute s = x + 3/2 and ds = dx.
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Factor -1/4 from the denominator:
 = (5 log(3))/2 + 1/2 integral_(3/2)^(5/2) 4/(4 s^2 - 1) ds

Factor out constants:
 = (5 log(3))/2 + 2 integral_(3/2)^(5/2) 1/(4 s^2 - 1) ds

Factor -1 from the denominator:
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For the integrand 1/(1 - 4 s^2), substitute p = 2 s and dp = 2 ds.
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Apply the fundamental theorem of calculus.
The antiderivative of 1/(1 - p^2) is tanh^(-1)(p):
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Which is equal to:

Answer:  = log(18)
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