Derivation rule of sum and difference of functions
Law: The derivative of the sum (difference) of two differentiable functions is equal to the sum (difference) of the derivatives of these two functions.
The formula can be written as: Where u and v are differentiable functions.
Example: know, look for
Answer:
Example: know, look for
Answer:
Derivation rule of function product quotient
Derivation rule of product of constant and function
Rule: When finding the derivative of the product of constant and differentiable function, the constant factor can be mentioned outside the derivative sign. The formula can be written as:
Example: know, look for
Answer:
Derivation rule of function product
Rule: The derivative of the product of two differentiable functions is equal to the derivative of the first factor multiplied by the second factor, plus the derivative of the first factor multiplied by the second factor. The formula can be written as:
Example: know, look for
Answer:
Note: When multiplying three functions, consider two of them as one item first.
Derivation rule of function quotient
Rule: The derivative of the quotient of two differentiable functions is equal to the product of the numerator derivative and denominator derivative minus the product of denominator derivative and numerator derivative, divided by the square of denominator derivative. The formula can be written as:
Example: know, look for
Answer:
The concept of indefinite integral
The concept of primitive function
It is known that function f(x) is a function defined in a certain interval. If the function F(x) exists, it can be found at any point in the interval.
dF'(x)=f(x)dx，
Then the function f(x) is called the original function of the function F(x) in this interval.
Example: sinx is the original function of cosx.
On the original function problem
What conditions does the function f(x) satisfy to ensure that its original function must exist? We will solve this problem later. If there are primitive functions, how many primitive functions are there?
We can clearly see that if the function f(x) is the original function of the function F(x),
That is f "(x) = f (x),
Then any function in the function family F(x)+C(C is an arbitrary constant) must be the original function of F(x).
Therefore, if the function f(x) has an original function, then its original function is infinite.
The concept of indefinite integral
All primitive functions of the function f(x) are called indefinite integrals of the function f(x),
Write it down.
From the above definition, we can know that if the function f(x) is the original function of the function f(x), then the indefinite integral of F(x) is a family of functions.
F(x)+C。
Namely: =F(x)+C
For example, ask questions
Answer: Because, so =
Properties of indefinite integral
1, the indefinite integral of the sum of functions is equal to the sum of indefinite integrals of each function;
Namely:
2. When solving indefinite integral, the constant factor that is not zero in the integrand function can be mentioned outside the integral symbol.
Namely:
The method of finding indefinite integral
Alternative method
Substitution method (1): If f(u) has the original function F(u) and u=g(x) is derivable, then F[g(x)] is the original function of f[g(x)]g'(x).
In other words, there is an alternative formula:
Example: Find
Answer: This integral can't be found in the basic integral table, so method of substitution should be used.
Let u=2x, then cos2x=cosu and du=2dx, so:
Method of substitution (2): Let x=g(t) be a monotone differentiable function, let g'(t)≠0, and let f[g(t)]g'(t) have the original function φ(t).
Then φ[g(x)] is the original function of f(x) (where g(x) is the inverse function of x=g(t)).
In other words, there is an alternative formula:
Example: Find
Solution: The difficulty of this integral is that it has roots, but we can exchange elements with trigonometric formulas.
Let x = asint (-π/2
On the question of alternative methods.
According to the derivative rule of compound function, the method of substitution of indefinite integral is obtained. We should choose methods according to concrete examples. Finding indefinite integral is not as regular as finding derivative. To skillfully find indefinite integral of a function, you have to do a lot of practice.
Partial integral
This method is obtained by using the derivative rule of the product of two functions.
Let the functions u=u(x) and v=v(x) have continuous derivatives. As we know, the derivative formula of the product of two functions is:
(uv)'=u'v+uv', move this item to get.
Uv'=(uv)'-u'v, and get the indefinite integral on both sides:
,
This is a partial integral formula.
Example: Find
Answer: It is not easy to get the result by substitution, so we use integration by parts.
Let u=x and dv=cosxdx, then substitute du=dx and v=sinx into the partial integral formula:
The problem of partial integration.
When using partial integration, you should choose U and dv properly, otherwise it is completely different. There are two points to consider when choosing u and dv:
(1)v should be easily available;
(2) easy to accumulate.