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1. Equivalent Infinitesimals#

Equivalent infinitesimal substitution is a technique in calculus used to simplify limit calculations, especially in the process of differentiation and indefinite integration. Two functions have a limit of zero at a certain point, and if the limit of their ratio at that point is 1, then these two functions are equivalent infinitesimals near that point. Here are some common formulas for equivalent infinitesimal substitutions, which hold true as $x \to 0$:

1. $\sin(x) \sim x$
2. $\tan(x) \sim x$
3. $\arcsin(x) \sim x$
4. $\arctan(x) \sim x$
5. $1 - \cos(x) \sim \frac{1}{2}x^2$
6. $\ln(1 + x) \sim x$
7. $e^x - 1 \sim x$
8. $(1 + x)^a - 1 \sim ax$, where $a$ is any constant
9. $\log_a(1 + x) \sim \frac{x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

The concept of equivalent infinitesimals can be used to replace complex expressions with simple ones when calculating limits, as long as these expressions behave similarly near the point in question. These substitutions are only valid when $x$ approaches 0, because only under this condition is the limit of their ratio equal to 1.

For example, if we want to calculate $\lim_{x \to 0} \frac{\sin(x)}{x}$, we can directly use the equivalent infinitesimal substitution $\sin(x) \sim x$, and obtain:

$\lim_{x \to 0} \frac{\sin(x)}{x} = \lim_{x \to 0} \frac{x}{x} = 1$

The use of equivalent infinitesimal substitutions can greatly simplify the process of calculating limits.

2. Integration Substitution Formulas#

Integration substitution formulas are a method in integral calculus that transforms complex integrals into simpler forms through variable substitutions. Here are some commonly used integration substitution techniques:

1. Algebraic Substitution:

   • When the integrand contains square roots of the form $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$, trigonometric substitution is usually used.

2. Trigonometric Substitution:

   • For $\sqrt{a^2 - x^2}$, let $x = a\sin(\theta)$.
   • For $\sqrt{a^2 + x^2}$, let $x = a\tan(\theta)$.
   • For $\sqrt{x^2 - a^2}$, let $x = a\sec(\theta)$.

3. Integration by Parts:

   • According to the product rule of integration, $\int u dv = uv - \int v du$.

4. Integration of Rational Functions:

   • For rational functions (a polynomial divided by another polynomial), partial fraction decomposition can be used.

5. Integration of Trigonometric Functions:

   • Use trigonometric identities to simplify the integrand, such as $\sin^2(x) + \cos^2(x) = 1$.

6. Integration of Exponential and Logarithmic Functions:

   • For functions of the form $a^x$, use the natural logarithm base $e$ for substitution, i.e., $a^x = e^{x\ln(a)}$.

7. Integration by Completing the Differential:

   • Make an appropriate substitution that relates the integrand to its derivative.

8. Substitution Rule:

   • For integrals of composite functions, use the $u$-substitution, i.e., let $u = g(x)$, and then calculate $du = g'(x)dx$.

9. Integration of Inverse Trigonometric Functions:

   • When the integrand has a form similar to the derivative of an inverse trigonometric function, the integral of the inverse trigonometric function can be directly used.

10. Standard Forms of Specific Integrals:

    • Some integrals have known standard forms, such as $\int \frac{1}{1 + x^2} dx = \arctan(x) + C$.

When performing variable substitutions, it is important to remember to substitute the differential term ($dx$) as well, to ensure the correctness of the integral. After the substitution, the integral may become simpler and can be solved using basic integral formulas or further techniques. After completing the integration, if necessary, the variables should be substituted back to the original variables.

3. Common Standard Forms of Specific Integrals#

The standard forms of specific integrals usually refer to the indefinite integrals of some common functions, which have general integral formulas. Here are some basic indefinite integral formulas:

1. Integration of Power Functions:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$

2. Integration of Exponential Functions:

$\int e^x \, dx = e^x + C$

$\int a^x \, dx = \frac{a^x}{\ln(a)} + C \quad (a > 0, a \neq 1)$

3. Integration of Logarithmic Functions:

$\int \frac{1}{x} \, dx = \ln|x| + C$

4. Integration of Trigonometric Functions:

$\int \sin(x) \, dx = -\cos(x) + C$

$\int \cos(x) \, dx = \sin(x) + C$

$\int \sec^2(x) \, dx = \tan(x) + C$

$\int \csc^2(x) \, dx = -\cot(x) + C$

$\int \sec(x)\tan(x) \, dx = \sec(x) + C$

$\int \csc(x)\cot(x) \, dx = -\csc(x) + C$

5. Integration of Inverse Trigonometric Functions:

$\int \frac{1}{\sqrt{1 - x^2}} \, dx = \arcsin(x) + C$

$\int \frac{-1}{\sqrt{1 - x^2}} \, dx = \arccos(x) + C$

$\int \frac{1}{1 + x^2} \, dx = \arctan(x) + C$

$\int \frac{-1}{1 + x^2} \, dx = arccot(x) + C$

6. Integration of Hyperbolic Functions:

$\int \sinh(x) \, dx = \cosh(x) + C$

$\int \cosh(x) \, dx = \sinh(x) + C$

7. Integration of Inverse Hyperbolic Functions:

$\int \frac{1}{\sqrt{x^2 + 1}} \, dx = \text{arsinh}(x) + C$

$\int \frac{1}{\sqrt{x^2 - 1}} \, dx = \text{arcosh}(x) + C \quad (x > 1)$

These formulas are the starting point for solving basic integral problems, but in practical applications, you may need to combine multiple integration techniques, such as substitution, integration by parts, partial fraction decomposition, etc., to solve more complex integral problems. Remember, the integration constant $C$ represents the uncertainty of the integral, and it always appears in indefinite integrals.