Step-by-step explanation:
Hey there!
The equation of a circle which passes through any point having centre (h,k) is;
In first blank ( x-_ ) keep "-2". In second blank (y - _ ) keep "4" and in last blank ( _ ) keep "36".
<em><u>Hope</u></em><em><u> </u></em><em><u>it helps</u></em><em><u>.</u></em><em><u>.</u></em>
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>
Explanation:
As his uniform is made up of tan or blue pants and a blue or white collared shirt.
So, there are possibly four combinations which are as follows:
- tan pants/blue shirt
- tan pants/white shirt
- blue pants/blue shirt
- blue pants/white shirt
As Benjamin carries an extra piece of white shirt. So, he has a little bit better than 25% chance of wearing his favorite combination.
So,
- Probability of getting tan pants = 2/4 = 1/2
- Probability of getting white shirt = 3/5
The probability of getting both can be computed by simply multiplying 2/4 and 3/5.
So,
- Probability of getting both = 1/2 × 3/5 = 3/10 ⇒ 30%
<em>Keywords: probability, chance</em>
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