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:
706.95in²
Step-by-step explanation:
This problem bothers on the mensuration of solid shapes, a cylinder
Step one
The trash can has a cylindrical shape
The distance from the center to the edge is the radius r= 15in
Step two
Area of top (circular ) = πr²
A= 3.142*15²
A= 3.142*225
A= 706.95in²
Answer to question 1 is D
answer to question 2 is A
Answer:
200,300,290,320,340
Step-by-step explanation:
If the mean of five numbers = 290
Then the total sum of the five numbers should be= 290*5
= 1450
an example of a set of five different numbers whose mean is 290
= 200,300,290,320,340
Summation of the above
=( 200+300+290+320+340)
= 1450
So the above set of numbers is a good example of a set of five different numbers whose mean is 290
Answer: C and B
Step-by-step explanation:
