Simple summary of indefinite integral question types

indefinite integral

Mathematics review notes for postgraduate entrance examination, used to review knowledge points, if there are any deficiencies, please point out, Thanks♪(･ω･)ﾉ

1 Primitive function/indefinite integral concept and properties

2 Existence theorem of primitive function

Understand: 1. Continuity can lead to existence of primitive functions, but existence of primitive functions cannot lead to continuation.

2. The relationship between the first type of discontinuity point, the second type of discontinuity point and the original function.

example:

Do it by definition of continuity

Since there are discontinuities of the first kind, there is no primitive function

It embodies why discontinuity can also have the original function

3 Basic formulas of indefinite integrals

You must memorize everything by heart (even recite U·ェ·U backwards)

Reference article link: http://t.csdn.cn/76fsC

4 Basic calculation of indefinite integral

The calculation of the indefinite integral is a difficult point. Before starting the calculation, you need to memorize the knowledge points:

1. Basic formulas for indefinite integrals
2. Derivative Formulas of Basic Elementary Functions
3. double angle formula
4. $(a\pm b{)}^3$ Expand the formula
5. Various trigonometric functions and their transformations (such as $ta{n}^{2x}\_+1=se{c}^{2}x,there\; istx=\frac{1}{safe\_\_}$wait)
6. auxiliary angle formula
7. Some tricks (table method, undetermined coefficient method, etc.)
8. Sum and difference product, product and difference formula

Next, we briefly introduce the types of frequently asked questions.

4.1 Triangular substitution type

4.2 Integration by parts

4.3 Rational function integration

4.3.1 Partial fraction method

Mainly to clarify the principle of decomposing the denominator of the partial fraction method

4.3.2 Adding items and subtracting items

The most frequently tested question types require focused training! ! !

4.4 Trigonometric rational integral (R(sinx, cosx) type)

4.4.1 Universal Substitution

4.4.2 Triangular deformation/division/substitution/compact differentiation

Commonly used methods require focused training! ! !

The flexible use of triangular deformation, take the same up and down with cos and then add dsin

4.5 Simple irrational function integrals

Usually, just replace the root sign part directly:

Of course, sometimes there is no need to replace the situation.

Note! Here is a must-remember conclusion: $\int \frac{1}{\sqrt{x}}=2\sqrt{x}+C=2\int d\sqrt{x}$

4.6 Indefinite integral techniques

4.6.1 Tabular method

One of them can be changed to 0 after continuous derivation, and the tabular method can be used.

4.6.2 Determinant method

A trigonometric function of the form multiplied by a logarithm.

Calculation method:
$$\frac{1}{a^2+b^2}\left|\begin{array}{cc}(e^x{)}^{\prime }& (sinx{)}^{\prime }\\ e^x& sin\end{array}\right|\text{\hspace{1em}( 1 )}$$
The original question can be regarded as$\int e^{ax}sininbxdx$form.

The answer is: $\frac{1}{1+1}[(e^x{)}^{\prime }\cdot sinx-(sinx{)}^{\prime }\cdot e^x]+C$

Note that eit must be written first; the trigonometric function part cannot be higher than1Second-rate

4.6.3 Undetermined coefficient method

Specifically, the denominator remains unchanged, just convert the numerator into A·(分母)+ B·(分母)', solve for A and B, and substitute into
$$Ax+Bln|Denominator|\_\_+After\; C$$
is brought in, it is the final answer.

Note: For details, refer to the "Three Computing" video lessons or refer to the answer P117

4.6.4 Auxiliary angle formula

4.6.5 Sum-difference product/product sum-difference

accumulation and difference

4.6.6 Substitution of "1"