Postgraduate entrance examination | advanced mathematics Chapter4 indefinite integral

One's deceased father grind | Advanced mathematics Chapter4 Indefinite integral

I. Definition

1. Primitive function — If $F^{\prime }(x)=f(x),F(x)be\; calledf(x)Primitive\; function$
2. Indefinite integral — set up $F(x)byf(x)A\; primitive\; function\; of,F(x)+Cbe\; calledf(x)The\; indefinite\; integral\; of$
3. Notes:
   1. if $f(x)\exists \Rightarrow \exists$ Countless primitive functions
   2. The difference between any two primitive functions is a constant
   3. if $f(x)continuity\Rightarrow \exists$ Primitive function

II. Indefinite integral tool

a. The basic formula

1. $\int kdx=kx+C$
2. power function
   1. $\int x^{a}dx=\frac{1}{a+1}{x}^{a+1}+C;(a\ne 1)$
   2. $\int \frac{1}{x}dx=\ln |x|+C$
3. Exponential function
   1. $\int a^{x}dx=\frac{a^x}{\ln a}+C;(a\ne 1)$
   2. $\int 1^{x}dx=x+C$
4. Trigonometric functions

   $\int \sin xdx=-\cos x+C$	$\int \cos xdx=\sin x+C$
   $\int \tan xdx=-\ln |\cos x|+C$	$\int \cot xdx=\ln |\sin x|+C$
   $\int \sec xdx=\ln |\sec x+\tan x|+C$	$\int \csc xdx=\ln |\csc x-\cot x|+C$
   $\int {\sec }^{2}xdx=\tan x+C$	$\int {\csc }^{2}xdx=-\cot x+C$
   $\int \sec (\tan x)dx=\sec x+C$	$\int \csc x\cot xdx=-\csc x+C$

5. Square sum square difference

   $\int \frac{1}{\sqrt{1-x^2}}dx=\arcsin x+C$	$\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin \frac{x}{a}+C$
   $\int \frac{1}{1+x^2}dx=\arctan x+C$	$\int \frac{1}{a^2+x^2}dx=\frac{1}{a}\arctan \frac{x}{a}+C$
   $\int \frac{1}{\sqrt{x^2+a^2}}dx=\ln (x+\sqrt{x^2+a^2})+C$	$\int \frac{1}{\sqrt{x^2-a^2}}dx=\ln (x+\sqrt{x^2-a^2})+C$
   $\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln |\frac{x-a}{x+a}|+C$	$\int \sqrt{a^2-x}dx=\frac{a^2}{2}\arcsin \frac{x}{a}+\frac{x}{2}\sqrt{a^2-x^2}+C$

b. Integral method

case1: The first kind of substitution integral method

case2: The second kind of transformation integral method

1. Irrational becomes rational

2. Triangular substitution , Square sum square difference

involves :
1. $\sqrt{a^2-x^2}\Rightarrow x=a\sin t\Rightarrow a\cos t$
2. $\sqrt{a^2+x^2}\Rightarrow x=a\tan t\Rightarrow a\sec t$
3. $\sqrt{x^2-a^2}\Rightarrow x=a\sec t\Rightarrow a\tan t$

3. Integration by parts

$$\begin{array}{rl}(uv{)}^{\prime }& =u^{\prime }v+u{v}^{\prime }\\ \int (uv{)}^{\prime }dx& =\int u^{\prime }vdx+\int u{v}^{\prime }dx\\ uv& =\int vdu+\int udv\\ \int udv& =uv-\int vdu\end{array}$$

1. $\int power\ast fingerdx$, Left power function
   

2. $\int power\ast logarithmdx$, Retention logarithm
   

3. $\int power\ast Triangledx$
   Notes:

   1. Trigonometric functions $\sin \cos$ It must be once
      
   2. If you encounter ${\sin }^2{\cos }^2$ Reduce the degree with half angle formula
      $$\begin{gathered} \sin (2x)=2\sin x\cos x \\ \cos (2x)={\cos }^{2}x+{\sin }^{2}x \end{gathered}$$
      
   3. If you encounter $\tan \cot \sec \cot$ Wait for an even number of times
      

4. $\int power\ast Anti\; triangledx$, Leave inverse trigonometric function
   
   

5. $\int e^{ax}\ast \sin bx$ or $\int e^{ax}\ast \cos bx$, Leave trigonometric function
   

6. $\int {\sec }^{n}xdx$ or $\int {\csc }^{n}xdx$, among $n$ It's odd
   

7. $\int {\sec }^{n}xdx$ or $\int {\csc }^{n}xdx$, among $n$ For the even
   

III. Indefinite integral of special function

a. Rational functions

Definition : Integrand function $R(x)$ Is a rational function , among $R(x)=\frac{P(x)}{Q(x)}$, among $P(x)Q(x)$ They are polynomials
When $P(x)Maximum\; number\; of\; times<Q(x)Maximum\; number\; of\; times$, $R(x)$ Is the true fraction
When $P(x)Maximum\; number\; of\; times\ge Q(x)Maximum\; number\; of\; times$, $R(x)$ Is a false fraction

case1: False fraction

Ideas : take False fraction Turn into polynomial + True fraction

case2: True fraction

Ideas : 1. Molecules do not move ; 2. Denominator factorization into partial sums

b. Irrational function

c. Indefinite integral of trigonometric rational function