(1) There are a total of 3 red marbles + 4 green marbles = 7 marbles in the bag. The probability of selecting a red marble is given by the number of red marbles divided by the total number of marbles. Therefore, the probability of selecting a red marble is 3/7.

(2) To solve the given system of equations:
2x - y = 5 --(1)
3x + y = 1 --(2)

We can add equation (1) and equation (2) to eliminate the variable y:
(2x - y) + (3x + y) = 5 + 1
5x = 6
x = 6/5

Substitute the value of x into either equation (1) or equation (2) to solve for y:
3(6/5) + y = 1
18/5 + y = 1
y = 1 - 18/5
y = -13/5

Therefore, the solution to the system of equations is x = 6/5 and y = -13/5.

(3) Let's denote the amount of chestnuts Peter picked as x kg. Mary then picked twice as much, which is 2x kg, and Lucy picked 2 kg more than Peter, so she picked x + 2 kg.

The total amount of chestnuts picked is x + 2x + x + 2 = 26 kg.

Combining like terms, we have 4x + 2 = 26.
Subtracting 2 from both sides, we get 4x = 24.
Dividing both sides by 4, we find x = 6.

Therefore, Peter picked 6 kg, Mary picked 2(6) = 12 kg, and Lucy picked 6 + 2 = 8 kg.

(4) Let's represent the number the student chose as n. The problem states that the student chose a number, multiplied it by 2, subtracted 138, and got 102. We can write this as an equation:

2n - 138 = 102.

Adding 138 to both sides, we have:

2n = 240.

Dividing both sides by 2, we find:

n = 120.

Therefore, the number the student chose is 120.

(5) The integral of x^2 can be found by using the power rule for integration. According to the power rule, the integral of x^n with respect to x is (x^(n+1))/(n+1) + C, where C is the constant of integration.

Applying the power rule to x^2, we have:

∫(x^2) dx = (x^(2+1))/(2+1) + C
= (x^3)/3 + C.

So, the integral of x^2 dx is (x^3)/3 + C (where C is the constant of integration).

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