Question

Вычислить 1)∫ (3-x)dx2)∫

Answer 1

Ответ:

Непосредственное интегрирование, табличные интегралы .

$1)\ \ \displaystyle \int (3-x)\, dx=3\int dx-\int x\, dx=3x-\frac{x^2}{2}+C\\\\\\2)\ \ \int (4x-x^2)\, dx=4\int x\, dx-\int x^2\, dx=4\cdot \frac{x^2}{2}-\frac{x^3}{3}+C=2x^2-\frac{x^3}{3}+C$

Answer 2

To calculate the integrals you provided, we can use the rules of integration. The integral of a function represents the area under the curve of the function.

1) ∫ (3 - x) dx:

To integrate this function, we can use the power rule of integration. The power rule states that the integral of x^n is (x^(n+1))/(n+1). In this case, the function is 3 - x, which can be written as 3 - x^1.

Applying the power rule, we get:

∫ (3 - x) dx = ∫ 3 dx - ∫ x dx

The integral of a constant is the constant multiplied by x, and the integral of x is (1/2)x^2. Therefore, we have:

∫ (3 - x) dx = 3x - (1/2)x^2 + C

where C is the constant of integration.

2) ∫ dx:

This integral represents the integral of the constant function 1, which is simply x. Therefore, we have:

∫ dx = x + C

where C is the constant of integration.

So, the solutions to the integrals are:

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

where C represents the constant of integration.