Note that x² + 2x + 3 = x² + x + 3 + x. So your integrand can be written as
<span>(x² + x + 3 + x)/(x² + x + 3) = 1 + x/(x² + x + 3). </span>
<span>Next, complete the square. </span>
<span>x² + x + 3 = x² + x + 1/4 + 11/4 = (x + 1/2)² + (√(11)/2)² </span>
<span>Also, for the x in the numerator </span>
<span>x = x + 1/2 - 1/2. </span>
<span>So </span>
<span>(x² + 2x + 3)/(x² + x + 3) = 1 + (x + 1/2)/[(x + 1/2)² + (√(11)/2)²] - 1/2/[(x + 1/2)² + (√(11)/2)²]. </span>
<span>Integrate term by term to get </span>
<span>∫ (x² + 2x + 3)/(x² + x + 3) dx = x + (1/2) ln(x² + x + 3) - (1/√(11)) arctan(2(x + 1/2)/√(11)) + C </span>
<span>b) Use the fact that ln(x) = 2 ln√(x). Then put u = √(x), du = 1/[2√(x)] dx. </span>
<span>∫ ln(x)/√(x) dx = 4 ∫ ln u du = 4 u ln(u) - u + C = 4√(x) ln√(x) - √(x) + C </span>
<span>= 2 √(x) ln(x) - √(x) + C. </span>
<span>c) There are different approaches to this. One is to multiply and divide by e^x, then use u = e^x. </span>
<span>∫ 1/(e^(-x) + e^x) dx = ∫ e^x/(1 + e^(2x)) dx = ∫ du/(1 + u²) = arctan(u) + C </span>
<span>= arctan(e^x) + C.</span>
Answer:
Explanation:
<u>1. Calculate the monthly interest owed during year 1</u>
<u />
- <em>Interest for first year: 8%</em>
- The monthly rate is the yearly rate divided by 12: 8% / 12 = 0.08/12
- The monthly interest owed is the monthly rate times the balance: (0.08/12)×$1,800 = $12.00
<u>2. Calculate the monthly interest owed during year 2</u>
<u />
- <em>Interest for second year: 23%</em>
- The montly rate is the yearly rate divided by 12: 23% / 12 = 0.23/12
- The monthly interest owed is the monthly rate times the balance: (0.23/12)×$1,800 = $34.50
<u>3. Calculate the difference</u>
- Difference in the monthly interest owed during year 1 and year 2 = $34.50 - $12.00 = $22.50
Hence, the answer is the option c) $22.50
Answer:
hello the correct answer is y=30x+20
Step-by-step explanation:
Answers:
y = 50
angle AOB = 100
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Explanation:
Angle x is an inscribed angle that subtends or cuts off minor arc AB. This is the shortest distance from A to B along the circle's edge.
Angle y is also an inscribed angle that cuts off the same minor arc AB. Therefore, it is the same measure as angle x. We can drag point D anywhere you want, and angle y will still be an inscribed angle and still be the same measure as x.
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Point O is the center of the circle. This is because "circle O" is named by its center point.
Angle AOB is considered a central angle as its vertex point is the center of the circle.
Because AOB cuts off minor arc AB, and it's a central angle, it must be twice that of the inscribed angle that cuts off the same arc.
This is the inscribed angle theorem.
Using this theorem, we can say the following
central angle = 2*(inscribed angle)
angle AOB = 2*(angle x)
angle AOB = 2*50
angle AOB = 100 degrees
Answer:
Step-by-step explanation:
47 and half .................
