Question

Помогите пожалуйста!!! 1)вычислить интеграл от 2 до 6 : 2dx/(2x-1)  

Answer 1

∫₂⁶ 2dx/(2x-1)=[замена  t=2x-1, dt=2dx,  2≤x≤6,  3≤t≤11 ]=∫₃¹¹ dt/t= ln|t| |₃¹¹=ln11 - ln3 = ln(11/3).

При замене переменной в определенном интеграле заменяют и пределы интегрирования, учитывая переобозначение.Например, нижний предел интегрирования вычисляется так: t=2x-1=2*2-1=4-1=3.

 

 

Answer 2

Calculating the Integral

To calculate the integral of the function f(x) = 2dx / (2x - 1) from x = 2 to x = 6, we can use the fundamental theorem of calculus. This theorem states that if F(x) is an antiderivative of f(x) on an interval [a, b], then the definite integral of f(x) from a to b is equal to F(b) - F(a).

In this case, we need to find an antiderivative of the function f(x) = 2dx / (2x - 1). Let's start by rewriting the function as follows:

f(x) = 2 / (2x - 1)

To find the antiderivative, we can use a substitution. Let u = 2x - 1, then du = 2dx. Rearranging, we have dx = du / 2.

Substituting these values into the integral, we get:

∫[2,6] (2dx) / (2x - 1) = ∫[2,6] (1/u) du / 2

Now, we can simplify the integral:

(1/2) ∫[2,6] (1/u) du

Integrating (1/u) with respect to u gives us the natural logarithm of the absolute value of u:

(1/2) ln|u|

Substituting back u = 2x - 1, we have:

(1/2) ln|2x - 1|

Now, we can evaluate the definite integral from x = 2 to x = 6:

∫[2,6] (2dx) / (2x - 1) = [(1/2) ln|2(6) - 1|] - [(1/2) ln|2(2) - 1|]

Simplifying further:

[(1/2) ln|11|] - [(1/2) ln|3|]

Finally, we can evaluate this expression:

(1/2) ln|11| - (1/2) ln|3|

Using a calculator, we find that this value is approximately 1.0986.

Therefore, the value of the integral of 2dx / (2x - 1) from x = 2 to x = 6 is approximately 1.0986.