Advanced Mathematics: Fifth Chapter definite integral (2) Change element integral method Generalized integral method of general integration _ higher Mathematics

The change method of §5.4 definite integral

One, the formula of the Exchange element

"Theorem" if

1, function on the continuous;

2. The function is single value on the interval and has continuous derivative;

3, when on the change in the value of the change, and

，

Then there are

(1)

Prove:

The integrand in (1) is continuous in its integral interval, so the definite integral at both ends of (1) type exists. and (1) The original function of the integrand at both ends of the formula is present.

Suppose to be a primitive function on the, according to the Newton-Leibniz formula there

On the other hand, the derivative of the function is

This shows that the function is a primitive function on the above, so there are:

thereby having

A few comments on this theorem are given:

1. Replace the original variable with the new variable, the limit of the original integral should be replaced by the new variable.

After the original function is calculated, it is not necessary to transform the function of the original variable as the indefinite integral, and then subtract the upper and lower bounds of the new variable.

2, should pay attention to the substitution condition, avoids the mistake.

(1), in a single value and continuous;

(2),

3. For the time being, the change formula (1) is still established.

"Example 1" asks

"Solution One" order

At that time, at that time,.

Also at that time, there

and the transformation function is on the single value, in the continuous,

By the formula for the exchange of elements have

"Solution Two" order

At that time, at that time,.

Also, at that time,

and the transformation function is on the single value, in the continuous,

By the formula for the exchange of elements have

Attention:

In "solution Two", after the change, the lower limit of the definite integral is larger than the upper limit.

The change formula can also be reversed, i.e.

"Example 2" asks

Solution: Set,

At that time, at that time,

Generally speaking, this kind of exchange can not obviously write the new variable, naturally also do not have to change the upper and lower limits of the definite integral.

Two, commonly used variable substitution technique and several commonly used conclusion

"Example 3" proves

1, if in the continuous and for even function, then

2, if on the continuous and for odd functions, then

It is proved that the additive of interval by definite integral can be

To make a substitute for

Therefore there are

If it is even function, then

If it is a singular function, the

"Example 4", if in continuous, proves:

1,

2,

And the definite integral is calculated by this formula

1, Proof: Set,

2, Proof: Set,

"Example 5" asks

Solution: Order,

So

Commentary:

The calculation of the definite integral does not seek the original function, only uses the variable substitution, the definite integral property, this solution is worth us to learn.

The partial integral method of §5.5 definite integral

To set a function that has a continuous guide function on the interval, then

and

So

This is the partial integral formula for definite integrals.

Can also be written in a form

"Example 1" asks

The solution: Order,

At that time, at that time,.

"Example 2" calculates the definite integral (the natural number).

Solution: Set,

，

In this way, we get the recurrence formula, according to this formula, then calculate two simple initial values and, can be obtained.

，

When an even number, there are

Introduction Mark

Similarly, if it is odd, there are

Synthesis then gets the famous Wallace Formula One

Because, therefore

"Example 3" (for natural number)

Solution: Order,

At that time, at that time,

"Example 4" (Wallace Formula II)

Proof: Set

At that time, there

If it is an even number, there is

If it is odd, the

To synthesize the famous and commonly used Wallace Formula Two

The application of Wallace formula is very extensive, so it can conveniently find many definite integrals.

"Example 5" asks

Solution: Applying the Formula Two, there

§5.7 Generalized integral

"Introduction" Calculates the area of the curved edge trapezoid surrounding the curve and the positive half of the shaft.

According to the geometric meaning of the definite integral, the curved edge trapezoid area should be.

Obviously, this integral is no longer an ordinary definite integral, because its integral limit is positive infinity.

How do you find the value of this "new definite integral"? First use a computer to do a numerical experiment:

The computed values are programmed and the images of these values are made to see if the image is approaching a fixed line.

Please run the MATLAB program GS0504.M.

A generalized integral with an infinite interval of integral interval

"Definition One"

Set the function to be contiguous on the interval, if the limit

exists, the limit value is defined as the generalized integral of the function in the infinite interval and is recorded as

At this time, also called the generalized integral convergence;

If the above limit does not exist, then the generalized integral divergence is called.

Similar to

Set the function to be contiguous on the interval, if the limit

exists, the limit is called the generalized integral of the function on the infinite interval,

To make a note of, i.e.

At this time, it is also called generalized integral convergence, if the above limit does not exist, then the generalized integral divergence is called.

Similar to

Set the function to be contiguous on the interval, if the generalized integral

And

At the same time convergence, the sum of the two generalized integrals is called the generalized integral of the function in the infinite interval.

i.e.

In this case, the generalized integral convergence is also called, if the above limit does not exist, then the generalized integral divergence is called.

The above integral is called the generalized integral of infinite limit.

"Counter Case"

But

Divergence, therefore, is divergent.

"Example 1" Computing generalized integrals

Solution:

Obviously, the infinite limit generalized integral is the limit of definite integral on any finite interval.

"Example 2" calculates the generalized integral.

Solution:

To observe the problem-solving process, the limit symbol will not participate in the operation until the last minute, for convenience, we can write it in the following form:

Please note: When the upper and lower limits are taken into the original function, it means the limit

This agreement, does not change the infinite limit generalized integral essence, but makes the mark concise many, and with the definite integral computation program basically is consistent.

"Example 3" shows that the generalized integrals converge at that time;

Solution: If

If

Generalized integrals of two unbounded functions

"Definition Two"

Set the function to be contiguous on the interval, and, take,

If the limit exists, the limit value is called the generalized integral of the function on the interval. i.e.

At this time, it is also called generalized integral convergence, if the above limit does not exist, then the generalized integral divergence is called. Point is called the Singularity Point.

Similarly, there

The function is contiguous on the interval, and, if the limit exists, the limit is called the generalized integral of the function on the interval. i.e.

At this time, it is also called generalized integral convergence, if the above limit does not exist, then the generalized integral divergence is called. Point is called the Singularity Point.

Similarly, there are

Set the function to be contiguous except on the top, and,

If two generalized integrals and all converge, then the generalized integral is defined

Otherwise the generalized integral divergence is called. Point is called the singularity.

Note: In the upper-style and