Question

Интеграл

Answer 1

Используем подстановку
u=tg(x/2)   du=(1/2)dx/(sin(x/2))^2, тогда
sinx=2u/(u^2+1)  cosx=(1-u^2)/(u^2+1) dx=2du/(u^2+1), получили
∫2du/((u^2+1)(2au/(u^2+1)+3(1-u^2)/(u^2+1)), упростив знаменатель, имеем
∫2du/(-3u^2+2au+3)=2∫du/(-3u^2+2au+3)=2∫(1/12)(36+4a^2)-(√3u-a/√3)^2)
s=√3u-pi/3     ds=√3du, получим
2/√3∫12ds/(36+4a^2)-s^2)=8√3/(36+4a^2)∫ds/(1-3s^2/(9+a^2))
Снова заменим p=√(3/(a^2+9)  dp=√((3/(a^2+9))ds
8√(9+a^2)/(36+4a^2)∫dp/(a-p^2)=8√(9+a^2)Arth(p)/(36+4a^2)+С
Делаем возврат
8√(9+a^2)Arth(p)/(36+4a^2)+С=2Arth(√(3s/(a^2+9))/(9+a^2)+C=2Arth(3u-a)/(√(9+a^2))/√(9+a^2)+C=2Arth(3tg(x/2)-a)/(√(9a^2))/√(9+a^2)+C

Answer 2

Understanding Integrals

An integral is a fundamental concept in calculus that represents the area under a curve. It is used to calculate quantities such as the area, volume, and accumulation of quantities over a given interval. Integrals have various applications in mathematics, physics, engineering, and other fields.

Evaluating Integrals

To evaluate an integral, you need to find an antiderivative (also known as an indefinite integral) of the function being integrated. The antiderivative of a function is a function whose derivative is equal to the original function. Once you find the antiderivative, you can evaluate the integral by subtracting the antiderivative at the upper limit from the antiderivative at the lower limit.

The general notation for evaluating an integral is:

∫ f(x) dx = F(x) + C

Where: - ∫ represents the integral symbol. - f(x) is the function being integrated. - dx represents the variable of integration. - F(x) is the antiderivative of f(x). - C is the constant of integration.

Example Integral

Let's consider the integral of the function (x^2+3x-4)*ln(x) with respect to x. Unfortunately, the search results did not provide the complete solution for this specific integral. However, I can guide you through the general process of evaluating integrals.

To evaluate the integral, you would need to find the antiderivative of the function (x^2+3x-4)*ln(x). This involves applying integration techniques such as substitution, integration by parts, or other methods depending on the complexity of the function.

Unfortunately, without further information or the complete solution, I am unable to provide the specific value of the integral. However, if you have any other questions or need assistance with a different integral, please let me know!