1. Understand the concepts of original function and indefinite integral and definite integral.

2. Master the basic formula of indefinite integral, the properties of indefinite integral and definite integral, the mean value theorem of definite integral, and the integration methods of substitution integral and partial integral.

3. Can find the integral of rational function, rational formula of trigonometric function and simple unreasonable function.

4. If you understand the function of the upper limit of integral, you will find its derivative and master Newton-Leibniz formula.

5. Understand the concept of generalized integral and calculate generalized integral.

6. Master the representation and calculation of some geometric and physical quantities (the area of plane figure, the arc length of plane curve, the volume and lateral area of rotating body, the area of parallel section is the volume, work, gravity, pressure, centroid, centroid, etc. of a known solid). ) and definite integral to find the average value of the function.

The key to simplify indefinite integral with the second kind of method of substitution is to choose a suitable transformation formula x = φ(t). This method is mainly to find the indefinite integral of irrational function (function with root sign). Because it is difficult to integrate the sum root, we try to eliminate the root by substitution to make it easy to calculate.

Let me briefly introduce the methods commonly used in the second alternative method:

(1) radical substitution: the integrand has the radical √(ax+b), which can directly make t = √ (ax+b);

(2) Trigonometric substitution: using trigonometric function substitution, variable root integral is rational function integral, and there are three kinds:

The integrand function contains the radical √ (a 2-x 2), so x = asint.

The integrand function contains the radical √ (a 2+x 2), so x = atant.

The integrand function contains the radical √ (x 2-a 2), so x = asect.

Note: Remember that triangular graphs can facilitate variable recovery.

There are several alternative forms:

(3) Inverse substitution (that is, x = 1/t): Let m and n be the highest degree of the numerator and denominator of the integrand function relative to x, respectively. When n-m > 1, inversion is expected to succeed;

(4) Exponential substitution: it is suitable for algebraic expressions in which the integrand function consists of exponent a x;

(5) Universal substitution (half-angle substitution): The integrand is a rational trigonometric function, which can make t = tan(x/2).