Definite integral for Advanced Mathematics: upper limit 1 lower limit 0 1 / (x ^ 2 + X + 1) DX I think I'm quite right, but it's different from the answer!

Definite integral for Advanced Mathematics: upper limit 1 lower limit 0 1 / (x ^ 2 + X + 1) DX I think I'm quite right, but it's different from the answer!

∫1/(x^2＋x＋1)dx
=∫1/[(x+1/2)^2+3/4]dx
=4/3*∫1/[(2√3x/3+√3/3)^2+1]*√3/2d(2√3x/3+√3/3)
=8√3/9*∫1/[(2√3x/3+√3/3)^2+1]d(2√3x/3+√3/3)
=8√3/9*arctan(2√3x/3+√3/3)
So the result of the definite integral is
8√3/9*arctan√3-8√3/9*arctan√3/3
=8√3/9*(π/3-π/6)
=4√3/27*π
I hope it will help the landlord,

F (x) = x / (1 + x) + √ (1-x) ∫ f (x) DX, then f (x) = how much. That's a definite integral, and the upper and lower limits of the integral are 1 and 0 respectively

four hundred and forty-four

The derivative and second derivative of y = f (x) are greater than 0, △ y = f (x + △ x) - f (x). When △ x is greater than 0, compare Dy and △ y

Because the derivative and second derivative of y = f (x) are greater than 0,
It is a monotonically increasing concave function
△y=f(x+△x)-f(x)
When △ x is greater than 0,
dy=f'(x)dx=f'(x)△x
Combined with image knowledge
△y>dy.

Why do we have to multiply DX / dy after the second derivative of the inverse function of a function

This kind of problem needs to distinguish who is the derivative object. DX / dy = 1 / y 'is the derivative formula of the inverse function. Both sides derive y at the same time, and the left = D ² x/dy ² And if you just write on the right: - y '' / (y ') ², At this time, the derivative of X on the right is inconsistent with the derivative object on the left, so it is wrong, because this question is

Let f (x) have continuous derivatives on interval [0,1], and prove that x ∈ [0,1] has | f (x) | ≤ ∫ < 0,1 > (| f (T) | + | f '(T) |) DT

0

How to find the definition domain of function in high school mathematics

In fact, the function definition field refers to the value range of X in function f (x). When doing a topic, first see what kind of function it is. The common special point is the logarithmic function f (x) = logx, so it is sure that x is greater than 0; Note that the denominator of the inverse proportional function is not 0. Then pay attention to whether the question has requirements for X, especially the application question