Chapter 4: Indefinite Integrals — Advanced Mathematics - Collection of Formulas

Chapter 4: Indefinite Integrals

4.1 Fundamentals of Indefinite Integrals

4.1.1 Basic Concepts

Nature 1:

• $(\int f(x)\mathrm{d}x)^{\prime}=f(x)$
• $\mathrm{d}\int f(x)\mathrm{d}x=f(x)\mathrm{d}x$

Property 2:

• $\int f^{\prime}(x)\operatorname{d}x=f(x)+C$
• $\int\operatorname{d}f(x)=f(x)+C$

Nature 3:

• $\int[f(x)\pm g(x)]\operatorname{d}x=\int f(x)\operatorname{d}x\pm\int g(x)\operatorname{d}x$

Property 4:

• $\int kf(x)\operatorname{d}x=k\int f(x)\operatorname{d}x$

4.1.2 Basic Formulas

Basic Formulas

$$1、\int adx=ax+C\:,\:a\text{ This is a constant }$$

$$2、\int x^{a}dx=\frac{x^{a+1}}{a+1}+C,\,\text{ where } a \text{ is a constant, and } a\neq-1$$

$$3、\int\frac{1}{x}dx=\ln|x|+C$$

$$4、\int e^{x}dx=e^{x}+C$$

$$5.\quad\int a^{x}dx=\frac{a^x}{\ln a}+C,\qquad a>0,\ a\neq1$$

$$6、\int\sin xdx=-\cos x+C$$

$$7、\int\cos xdx=\sin x+C$$

$$8、\int\sec^{2}xdx=\tan x+C$$

$$9、\int\csc^{2}xdx=-\cot x+C$$

$$10、\int\tan xdx=-\ln|\cos x|+C$$

$$11、\int\cot xdx=\ln\lvert\sin x\rvert+C$$

$$12、\int\sec xdx=\ln\lvert\sec x+\tan x\rvert+C$$

$$13、\int\csc xdx=-\ln\lvert\csc x+\cot x\rvert+C$$

$$14、\int\frac{dx}{1+x^2}=\arctan x+C$$

$$15、\int\frac1{\sqrt{a^2-x^2}}dx=\arcsin\frac xa+C$$

$$16、\int\frac1{x^2-a^2}dx=\frac1{2a}\ln\left|\frac{x-a}{x+a}\right|+C$$

$$17、\int\frac{dx}{\sqrt{1-x^2}}=\arcsin x+C$$

$$18、 \int\frac 1{a^2+x^2}dx=\frac 1 a\arctan\frac xa+C$$

$$19、\int\frac{dx}{\sqrt{x^2+a^2}}=\ln (x+\sqrt{x^2+a^2})+C$$

$$20、\int\frac{dx}{\sqrt{x^2-a^2}}=\ln\left|x+\sqrt{x^2-a^2}\right.|+C$$

Trigonometric Function Collection

Common Trigonometric Integrals

• Corresponding one-to-one with differentiation:
• $$\int\sin xdx=-\cos x+C、\int\cos xdx=\sin x+C$$
• $$\int\tan xdx=-\ln|\cos x|+C、\int\cot xdx=\ln\lvert\sin x\rvert+C$$
• $$\int\sec xdx=\ln\lvert\sec x+\tan x\rvert+C、\int\csc xdx=-\ln\lvert\csc x+\cot x\rvert+C$$
• $$\int\sec^{2}xdx=\tan x+C、\int\csc^{2}xdx=-\cot x+C$$
• $$\int\tan x secx=\sec x+C、\int{\cot x\,\csc x}\,dx=-csc x+C$$

Common Inverse Trigonometric Integrals

• $$\int\frac{dx}{\sqrt{1-x^2}}=\arcsin x+C$$
• $$\int\frac1{\sqrt{a^2-x^2}}dx=\arcsin\frac xa+C$$
• $$\int\frac{dx}{1+x^2}=\arctan x+C$$
• $$\int\frac 1{a^2+x^2}dx=\frac 1 a\arctan\frac xa+C$$

4.2 Solving Indefinite Integrals

4.2.1 Method One: First Type of Substitution Integral

First Type of Substitution Integral Method

• Definition:
• Also known as the convergent differential method;
• If $\int f(u)\mathrm{d}u=F(u)+C$, $\text{ then }\int f[\varphi(x)]\varphi^{\prime}(x)\operatorname{d}x=\int f[\varphi(x)]\operatorname{d}\varphi(x)=F[\varphi(x)]+C$

Summary of Differential Forms for Combining Differentials: Common functions

• 1. $$\int f( ax+ b) dx= \frac 1a\int f( ax+ b)d( ax+ b)$$
• 2. $$\int x^mf( ax^{m+ 1}+ b)dx=\frac 1{( m+ 1) a}\int f( ax^{m+ 1}+ b)d(ax^{m+1}+ b)\quad\quad\quad ( m\neq- 1) $$
• 3. $$\int f( \sqrt {x}) \frac {\mathrm{d} x}{\sqrt {x}}= 2\int f( \sqrt {x})d( \sqrt x)$$
• 4. $$\int f( e^x) \mathrm{e} ^xdx= \int f( \mathrm{e} ^x)d(\mathrm{e} ^x)$$
• 5. $$\int f(\ln x)\:\frac{1}{x}\mathrm{d}x=\int f(\ln x)\mathrm{d}(\ln x)$$

Summary of Differential Forms: Common Trigonometric Functions

• 1. $$\int f(\sin x)\cos\:x\mathrm{d}x=\int f(\sin x)\mathrm{d}(\sin x)$$
• 2. $${\int}f(\cos x)\sin x\mathrm{d}x=-{\int}f(\cos x)\mathrm{d}(\cos x)$$
• 3. $$\int f(\tan x)\:\frac{1}{\cos^{2}x}\mathrm{d}x=\int f(\tan x)\mathrm{d}(\tan x)$$
• 4. $${\int}f(\arcsin x)\:\frac{1}{\sqrt{1-x^2}}\mathrm{d}x=\int f(\arcsin x)\mathrm{d}(\arcsin x)$$
• 5. $${\int}f(\arctan x)\:\frac1{1+x^2}\mathrm{d}x=\int f(\arctan x)\mathrm{d}(\arctan x)$$

4.2.2 Method Two: Second Type of Substitution Integral

Second Type of Substitution Integral Method

• Definition:
• $$\int f(x)\mathrm{d}x=\int f[\varphi(t)]\varphi^{\prime}(t)\mathrm{d}t=F(t)+C=F[\varphi^{-1}(x)]+C$$
• Method:
• The key is choosing a substitution: express $x$ in terms of a new variable, integrate, and then substitute back.

Form Summary

• The following three forms: $$\begin{aligned}&\sqrt{a^2-x^2} \\&\sqrt{a^2+x^2} \\&\sqrt{x^2-a^2}\end{aligned}$$
• Standard substitutions are: $$\begin{aligned}x&=a\sin t&&\text{for }\sqrt{a^2-x^2},\\\\x&=a\tan t&&\text{for }\sqrt{a^2+x^2},\\\\x&=a\sec t&&\text{for }\sqrt{x^2-a^2}.\end{aligned}$$

• Purpose: eliminate the radical and simplify the integrand.

4.2.3 Method 3: Integration by Parts

Integration by Parts

• Definition:
• Let $u(x),\nu(x)$ have consecutive first-order derivatives, then $\int udv=uv-\int vdu$
• Method:
• Suitable formultiplying two different functions;

Usage Scenarios

• Multinomial function × Exponent|Trigonometry:
• 1. $\int p_n(x)e^{ax}\operatorname{d}x$
• Score the index
• 2. $\int p_n(x)\sin ax\operatorname{d}x$
• Triangle Together
• 3. $\int p_n(x)\cos axdx$
• Triangle Together
• Multinomial Functions × Logarithm | Inverse Triangle:
• 4. $\int P_n(x)\ln xdx$
• Fill in polynomials
• 5. $\int P_n(x)\arctan xdx$
• Fill in polynomials
• 6. $\int P_n(x)\arcsin xdx$
• Fill in polynomials
• Exponential × Trigonometric Function
• 7. $\int e^{\alpha x}\sin\beta xdx$
• Either factor can be chosen first when applying integration by parts.
• 8. $\int e^{\alpha x}\cos\beta xdx$
• Either factor can be chosen first when applying integration by parts.