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5️⃣ Let A = {1, 2, 3} and B = {3, 4, 5}. Which of the following represents A - B?
4️⃣ Let A and B be two sets. Which of the following is true?
2️⃣If A = {x | x is an odd number less than 10}, what is A?
2️⃣0️⃣Evaluate the definite integral
∫[0, π] x^2 sin(x) dx.[‼️Hint: Use integration by Parts]
1️⃣9️⃣Evaluate the definite integral
∫[1, 4] (2x + 1)^2 dx.
1️⃣7️⃣Evaluate the definite integral
∫[0, π/2] sin(x) dx.
1️⃣6️⃣Evaluate the definite integral
∫[0, 1] (2x + 1) dx.
Repost from ExitExam+ESSLCE Support
✅️Solution 1️⃣4️⃣👆👆
Use 👉∫x^ndx=(x^{n+1})/(n+1)
then taking the integral of each term separately gives:
👉∫3x^2 dx = (3x^3)/3=x^3,
👉∫4x dx = (4x^2)/2=2x^2
👉∫1 dx = x.
Putting it all together, we get:
∫(3x^2 + 4x + 1) dx
= x^3 + 2x^2 + x + C.
Therefore, the correct answer is a) x^3 + 2x^2 + x + C.
✅Solution1️⃣5️⃣👆👆
From basic integrals(antiderivatives), we have: 👉the integral of e^x is e^x, and
👉the integral of 1/x is 5ln(x), so
∫2e^xdx =2∫e^xdx=2e^x, and
∫5/xdx=5 ∫1/xdx=5ln(x),
Putting it all together, we have:
∫(2e^x - 5/x) dx = 2e^x - 5ln(x) + C.
Therefore, the correct answer is a) 2e^x - 5ln(x) + C.
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✅️Solution 1️⃣4️⃣👆👆
Use 👉∫x^ndx=(x^{n-1})/n,
then taking the integral of each term separately gives:
👉∫3x^2 dx = (3x^3)/3=x^3,
👉∫4x dx = (4x^2)/2=2x^2
👉∫1 dx = x.
Putting it all together, we get:
∫(3x^2 + 4x + 1) dx
= x^3 + 2x^2 + x + C.
Therefore, the correct answer is a) x^3 + 2x^2 + x + C.
📝Solution 1️⃣3️⃣👆👆
📌If we apply the limit for numerator and denominator functions we get zero over zero, which is undefined.
📖However, if we first try simplifying we may get a defined result.
👉Now let us factor the numerator using the difference of squares formula as follows:
x^2 - 16 = (x - 4)(x + 4).
Then, canceling out the common factor of (x - 4) in the numerator and denominator, we get:
lim(x->4) [(x^2 - 16) / (x - 4) ]
= lim(x->4) (x + 4)
= 4 + 4
= 8.
Therefore, the correct answer is c) 8.
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1️⃣3️⃣Find the limit as x approaches 4 of (x^2 - 16) / (x - 4).
1️⃣3️⃣Find the limit as x approaches 4 of (x^2 - 16) / (x - 4).
📝Solution1️⃣1️⃣👆👆
We can factor the numerator using the sum of cubes formula:
x^3 + 8 = (x + 2)(x^2 - 2x + 4).
Canceling out the common factor of (x + 2) in the numerator and denominator, we have:
lim(x-> -2) (x^3 + 8) / (x + 2) = lim(x-> -2) (x^2 - 2x + 4)
= (-2)^2 - 2(-2) + 4
= 4 + 4 + 4
= 12.
📚Therefore, the answer is c) 12.
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1️⃣1️⃣Find the limit as x approaches -2 of (x^3 + 8) / (x + 2).
