Let f (x) = e ^ 2x, then the indefinite integral ∫ f '(x) DX is equal to the solution,

∫f'(x)dx
=f(x)+C
=e^2x+C

Indefinite integral, integrand function f (x) and DX, if f (x) becomes f (2x), DX also becomes D (2x), why do these two X have to be equal to calculate?

1. This is the correspondence emphasized in calculus, and the coresponding emphasized is the complete equivalence of independent variables;
2. If the integrand function sin2x of the integral, its independent variable is x, but the overall variable of the sine function is 2x,
If we integrate, the integrand is sin2x, but D is followed by X, that is to say, the value of the independent variable is the same as the integrand
The value of the product function is not in the same position. ∫ f (x) DX, where f (x) is the value of the function at position x, that is to say
Is the height of the graph, there is a rectangle at the position x, and the bottom width is DX
If ∫ f (2x) DX, it means that the bottom width at x is DX, but the height of the rectangle is 2x,
In this way, the basic meaning of integral is lost
3. On the whole, the height of the rectangle is f (2x), and the bottom width of the rectangle must be d (2x), that is, the function f must take a value at 2x,
At this time, 2x is understood as the original abscissa is compressed uniformly
When using the method of integration, the problem is the most common mistake for beginners

How to integrate the product of two functions
What's the answer upstairs?

The question of the landlord is too difficult to answer. It almost covers the whole integral theory. Examples are as follows:
1. The integral of xlnx needs the method of integration by parts;
2. The integral of (e ^ x) SiNx needs not only partial integral, but also integral equation;
3. The integral of 1 / (1 + X & # 178;) ^ n needs not only variable substitution, but also integral recursion and partial integral;
4. The integral of (SiNx) lnsinx not only needs to give the integral interval, but also needs to use the integral method of complex variable function;
、、、、、、、、、、、、、、
The problem of the landlord is a small one. It seems that the "method" is enough. After careful analysis, this problem can be solved
Including all aspects of integral. A Book of heaven is endless

The number of students in a class is between 40 and 50. If the class is divided into a group of 8 students, the number of students in a group is less than 3. If the class is divided into a group of 12 students,
Then one less in each of the three groups, ask for the total number of students in this class

In fact, if you add three people, it's just the common multiple of 8 and 12. The common multiple of 8 and 12 between 40 and 50 is 48, so the original number is 45

If a matrix is positive definite, does it have to be real symmetric?
If the premise is this, then the proposition "a, B are positive definite, then AB is also positive definite" does not hold?
Because if
A=2 1
1 3
B=2 1
1 2
AB=5 4
5 7
If AB is not a symmetric matrix, it is not positive definite

The premise is that the matrix must be real symmetric
therefore
A. If B is positive definite, then AB may not be positive definite

A necessary and sufficient condition for the product of complex number 1 + I and a + 1 to be a real number

(1+i)(a+1)
=a+1+(a+1)i
It's a real number
Then a + 1 = 0
a=-1
If a = - 1 (1 + I) * 0 = 0 is a real number
So if and only if a = - 1

Three teachers will take 38 students from class 41 to visit the science and Technology Museum. There are two kinds of vehicles available for rent: one is minivan, which can take up to seven people; the other is car
2. The rent for a minibus is 100 yuan per day, and the rent for a car is 120 yuan per day. How to rent a car at least? How much is it?

Six minivans for 600 yuan

How to write LIM (x → 0) (SiNx ^ 2) / (2x ^ 3)

Well, we can only analyze that when it is close to 0, it is close to 1 above and 0 below, so it is infinite

The even function f (x) defined on [- 2,2] is a monotone increasing function on [- 2,0]. If f (1-m) < f (m), the value range of M is obtained
Search for several answers, the process did not understand Bonus

Even functions defined on [- 2,2]
F (x) is a monotone increasing function on [- 2,0]
So the farther the function argument x is from the Y axis, the smaller the function value is
f(1-m)|m|
So - 1 ≤ m ≤ 3, - 2 ≤ m ≤ 2, M

How about a dress on me

How about I 'm wearing the clothes?

Let f (x) = e ^ 2x, then the indefinite integral ∫ f '(x) DX is equal to the solution,