Lecture 32: Calculation of definite integrals

1.1 Calculation of Definite Integrals

1.1.1 Concept Introduction

The Relationship Between Position Function and Velocity Function in Variable Speed Linear Motion

Defining definite integrals, which is cumbersome to use directly
$\int_a^bf(x)\operatorname{d}x\mathop{\Delta}\lim_{\lambda\to0}\sum_{i=1}^nf(\xi_i)\Delta x_i$
For example, when calculating distance, $v(t)$ is the velocity function, $s(t)$ is the distance function:
$\int_{T_1}^{T_2}\nu(t)dt=s(T_2)-s(T_1)\quad s^{\prime}(t)=\nu(t)$
And, according to the property of the indefinite integral: $s(t)=\int_{t_0}^tv(t)dt\quad s^{\prime}(t)=v(t)$
Can it be transformed into the difference between two points of the antiderivative of the integrand?

Case Analysis:Suppose $f(x)$ is continuous on $[a,b]$, and x is any point on $[a,b]$:

The area of the shaded part is called the function of theupper limit of integrals

$\Phi(x)=\int_a^xf(t)\mathrm{d}t\quad x\in[a,b]$

Question:Is this function with the upper limit of integration differentiable over the function interval?

Answer: Yes, provided $f(x)$ is continuous on $[a,b]$;

1.1.2 Functions and Derivatives of the Upper Limit of Integrals

#####Theorem: Afunctionoftheupperlimitoftheintegral: Fundamental theorem of calculus

Description: If $f(x)$ is continuity on $[a, b]$, then: $\int_a^xf(t)\mathrm{d}t$ is differentiable on $[a, b]$, and there is $$(\int_a^xf(t)\mathrm{d}t)^{\prime}=f(x)$$

Explanation

Concept:

-Derivative of the integral of a function, equal to the function value at this point;

When x is constant, it yields a fixed value;
Core:
Differentiation and integration are inverse operations;
Solves the existence of the antiderivative;
Conclusion:
$$(\int_{\varphi(x)}^{\psi(x)}f(t)dt)^{\prime}=f(\psi(x))\psi^{\prime}(x)-f(\varphi(x))\varphi^{\prime}(x)$$

#####Theorem: Parityoftheupperlimitfunctionoftheintegral

description:

1. If f(x) is an odd function, then $\int_0^xf(t)dt$ is even;

2. If f(x) is an even function, then $\int_0^xf(t)dt$ is an odd function;

Explanation

#####Theorem: Theoriginalfunctionofthedefiniteintegral

Description: If $f$ is continuous on $[a,b]$, then $F(x)=\int_a^x f(t)\,dt$ is an antiderivative of $f$ on $[a,b]$.

Explanation

When $f$ is continuous, the integral with variable upper limit, $F(x)=\int_a^x f(t)\,dt$, satisfies $F^{\prime}(x)=f(x)$.
Thus every continuous function has an antiderivative; the variable-upper-limit integral provides one.

Corollary: A more general form

If $\varphi$ and $\psi$ are differentiable and $f$ is continuous, then:
$\frac d{dx}\int_{\psi(x)}^{\varphi(x)}f(t)dt=f(\varphi(x))\varphi^{\prime}(x)-f(\psi(x))\psi^{\prime}(x)$

1.1.3 Example Problems

Example: $\text{Let }\Phi(x)=\int_{0}^{x^{2}}e^{-t^{2}}dt.\text{ Find }\Phi^{\prime}(x).$

Analysis
Apply the Fundamental Theorem of Calculus together with the chain rule.
Solution
$\Phi=\int_{0}^{u}e^{-t^{2}}dt$
Let $u=x^2$.
Differentiate the outer integral with respect to $u$, then multiply by $du/dx$:
$\bar{\Phi}_{x}^{\prime}(x)=\bar{\Phi}_{u}^{\prime}\cdot U_{x}^{\prime}=e^{-u^2}\cdot2x=e^{-x^4}\cdot2x.$
Question Type: Theoriginalfunctionofthedefiniteintegral

Example: When both upper and lower limits of the function change: $g(x)=\int_{\sin x}^{x^2}e^{-t^2}dt\quad$, find its derivative;

Analysis
It can be split into two segments, from sinx to 0, and then from 0 to x squared;
Solve separately and add them together;
Analysis
${\int_0^{x^2}e^{-t^2}dt+\int_{sinx}^0e^{-t^2}dt}$
Question Type: Theoriginalfunctionofthedefiniteintegral

1.2 Calculation Methods

1.2.1 Newton-Leibniz Formula

#####Theorem: NewtonLeibnizformula

Description: Let $F(x)$ be an antiderivative of continuous function $f(x)$ on $[a, b]$, then: $\int_{a}^{b}f(x)dx=F(b)-F(a)$, also called the basic formula of calculus;

Explanation

Turning the problem of solving definite integrals into the problem of finding the antiderivative;
And this formula communicates the relationship between integral calculus and differential calculus;

Corollary:Prove the mean value theorem of integrals n

If $f(x)$ has continuity on $[a, b]$, then
$\int_a^bf(x)\operatorname{d}x=f(\xi)(b-a)\quad\quad a<\xi<b$

1.2.2 Substitution Method

#####Theorem: Substitutionmethodfordefiniteintegrals

description: $$\int_{a}^{b}f(x)\operatorname{d}x=\int_{\alpha}^{\beta}f(\varphi(t))\varphi^{\prime}(t)\operatorname{d}t$$

Explanation

Note: The integrand and the upper and lower limits must be substituted;

1.2.3 Integration by Parts

#####Theorem: IntegrationbyPartsofDefiniteIntegrals

description: $$\int_{a}^{b}u\operatorname{d}v=uv\bigg|_{a}^{b}-\int_{a}^{b}v\operatorname{d}u$$

1.2.4 Other Methods

Method One: Utilizing Parity and Periodicity

$$\int_{-a}^af(x)\operatorname{d}x=\begin{cases}0,&f(x)&\text{ is the odd function, }\\2\int_0^af(x)\operatorname{d}x,&f(x)&\text{ is an even function }.\end{cases}$$
$$\int_a^{a+T}f(x)\operatorname{d}x=\int_0^Tf(x)\operatorname{d}x.$$

Method 2: Use Existing Formulas

$$\begin{aligned}(1)&\int_0^{\frac\pi2}\sin^nx\operatorname{d}x=\int_0^{\frac\pi2}\cos^nx\operatorname{d}x=\begin{cases}\frac{n-1}n\frac{n-3}{n2}\cdots\frac12\frac\pi2,&n\text{ Sorry }\\\frac{n-1}n\frac{n-3}{n-2}\cdots\frac23,&n\text{ Strange }&\end{cases}\\(2)&\int_0^{\pi}x\cdot f(\sin x)\operatorname{d}x=\frac\pi2\int_0^{\pi}f(\sin x)\operatorname{d}x\end{aligned}$$

1.2.2 Example Problems

Example Question: $\int_{0}^{\pi}\sin xdx$

Analysis
Analysis
$\int_{0}^{\pi}\sin xdx=\left(-cosx\right)\int_{0}^{\pi}=1-(-1)=2.$
Question Type: NewtonLeibnizformula

1.3 Frequently Tested Question Types

Question Type: Calculationofdefiniteintegrals

PART 1: Problem-solving methods

Five Common Methods

1. Newton-Leibniz formula;
2. Cycle change method
3. Partial Integration Method;
4. Parity, periodicity;
Obtain the function and first observe its parity and periodicity;
5. Two special formulas;
$$\begin{aligned}(1)&\int_0^{\frac\pi2}\sin^nx\operatorname{d}x=\int_0^{\frac\pi2}\cos^nx\operatorname{d}x=\begin{cases}\frac{n-1}n\frac{n-3}{n2}\cdots\frac12\frac\pi2,&n\text{ Sorry }\\\frac{n-1}n\frac{n-3}{n-2}\cdots\frac23,&n\text{ Strange }&\end{cases}\\(2)&\int_0^{\pi}x\cdot f(\sin x)\operatorname{d}x=\frac\pi2\int_0^{\pi}f(\sin x)\operatorname{d}x\end{aligned}$$
Supplementary Methods:
Use geometric analysis: If a function is a type of geometric figure, you can draw its shape and observe its geometric properties;

Common Conclusions in Indefinite Integral Geometry

1. $$\int_{0}^{a}\sqrt{a^{2}-x^{2}}dx=\frac{\pi a^{2}}{4}(a>0)$$
2. $$\int_{0}^{a}\sqrt{2ax-x^{2}}dx=\frac{\pi}{4}a^{2}$$
3. $$\int_{0}^{2a}\sqrt{2ax-x^{2}}dx=\frac{\pi}{2}a^{2}$$

Fixed Points with Variable Limits

When calculating definite integrals with variable upper limit functions, consider using the integration by parts method;

PART 2: Typical Example Problems

PART 3: Key Points Review

Question Type: Upgradedlimitedpoints

PART 1: Problem-solving methods

Core Formulas

Concept:
$$(\int_a^xf(t)\mathrm{d}t)^{\prime}=f(x)$$
Calculation:
$$(\int_{\varphi(x)}^{\psi(x)}f(t)dt)^{\prime}=f(\psi(x))\psi^{\prime}(x)-f(\varphi(x))\varphi^{\prime}(x)$$

Three Major Methods to Limit Points for Upgrades

Method 1: Formula calculation
Example: $$(\int_{e^{x}}^{x^{2}}f(t)dt)^{\prime}=f(x^{2})\cdot2x-f(e^{x})e^{x}$$
Method 2: Extract x
Example: $$\int_{0}^{x}(x-t)f(t)dt;=x\int_{0}^{x}f(t)dt-\int_{0}^{x}tf(t)dt$$
The integral domain and the integrand contain x;
When x can be proposed: split the terms and extract x;
Method 3: Substitution method
Example: $$\int_{0}^{x}cos(x-t)^{2}dt\frac{x-t=u}{}\int_{0}^{x}cos u^{2}du$$
The integral domain and the integrand contain x;
When x cannot be proposed: substitute;
Note: When substituting a substitution, x is a constant, and t and u are the integrand variables;
Example: When there is no x in both upper and lower limits, but x is present in the integral, you need to replace x and substitute it into the upper and lower limits;
After $x+t=u$: $$\int_{1}^{2}f(x+t)dt=\int_{x+1}^{x+2}f(u)du$$

Question Type: Variable Upper Limit Integral + Limit Calculation

When a limit contains a variable-upper-limit integral in the numerator or denominator, the Fundamental Theorem of Calculus often makes L'Hôpital's rule convenient.

Problem Type: Variable Upper Limit Integral + Differentiation of Implicit Functions

Example: Let the differentiable function $y=y(x)$ be determined by equation $\int_0^{x+y}\mathbf{e}^{-t^2}\mathbf{d}t=\int_0^xx\sin t^2\operatorname{d}t$, find $\frac{\operatorname{d}y}{\operatorname{d}x}_{x=0}$
Analysis: The left side of the equivalence is a function about x and y, and the right side of the equation is a function about x. Therefore, when you need to find the derivative of $y=y(x)$ on $\frac{\operatorname{d}y}{\operatorname{d}x}_{x=0}$, essentially it is an implicit function with y and x whose derivative is at point 0, so it is a problem of derivative of implicit functions;
Core: Derivative by the upper limit