Integral Calculus Compact Note

math-1109 MATH-1109 Differential and Integral Calculus

Comprehensive Note on Integration

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I. Indefinite Integration Techniques (Standard Forms and Substitutions)

A. Integrals Involving Linear and Quadratic Terms Under a Radical

Form	Suggested Substitution	Source Examples/Type
$\int \frac{dx}{\sqrt{(x-\alpha )(\beta -x)}}$	Let $x-\alpha =z^2$	Type $\frac{dx}{\sqrt{(\text{linear})(\text{linear})}}$ where factors are $(x-\alpha )$ and $(\beta -x)$.
$\int \frac{dx}{(ax+b)\sqrt{cx+d}}$	Let $cx+d=z^2$	Type $\frac{dx}{(\text{linear})\sqrt{\text{linear}}}$.
$\int \frac{dx}{a{x}^2+bx+c\sqrt{\beta x+\alpha }}$	Let $\beta x+\alpha =z^2$	Type $\frac{dx}{(\text{quadratic})\sqrt{\text{linear}}}$.

B. Integrals Involving Rational Functions of Trigonometry

1. Tangent Half-Angle Substitution (Universal Substitution): For integrals of the form $\int \frac{dx}{a+b\sin x}$ or similar forms involving $\sin x$ or $\cos x$ in the denominator, the substitution involves expressing trigonometric functions in terms of $t=\tan (x/2)$.

   • Example: $\int \frac{dx}{5-13\sin x}$.
   • Substitution: Let $\tan \frac{x}{2}=z$. This implies $dx=\frac{2dz}{1+z^2}$ and $\sin x=\frac{2z}{1+z^2}$.

2. Derivative-Based Decomposition (Linear/Linear Trigonometric Form): For integrals of the form $\int \frac{A\sin x+B\cos x}{C\sin x+D\cos x}dx$, the numerator is expressed as a linear combination of the denominator and its derivative: $\text{Numerator}=m(\text{Denominator})+n(\text{Derivative of Denominator})$

   • Example: $\int \frac{2\sin x+3\cos x}{3\sin x+4\cos x}dx$.
   • By equating coefficients, one finds $m=\frac{18}{25}$ and $n=\frac{1}{25}$. This allows the integral to be split into two parts: a simple integration of $m\int dx$ and an integral of the form $n\int \frac{f^{\prime }(x)}{f(x)}dx$, resulting in $n\ln |f(x)|$.

C. Other Suggested Substitutions

• Type $\int \frac{dx}{Px+q\sqrt{a{x}^2+bx+c}}$: Suggested substitution: Let $Px+q=\frac{1}{z}$.
• Type $\int \frac{dx}{(P{x}^2+q)\sqrt{a{x}^2+b}}$: Suggested substitution: Let $x=\frac{1}{z}$.

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II. Definite Integration

A. General Properties of Definite Integral

1. The value of a definite integral is independent of the variable of integration: ${\int }_a^{b}f(x)dx={\int }_a^{b}f(z)dz$
2. Reversing the limits of integration changes the sign of the integral: ${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$
3. The integral can be split over an intermediate point $c$ ($a\le c\le b$): ${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$
4. The property ${\int }_0^{a}f(x)dx$ can be transformed using the difference between the limit and the variable: ${\int }_0^{a}f(x)dx={\int }_0^{a}f(a-x)dx$
5. If $f(T+x)=f(x)$ (periodicity), then: ${\int }_0^{na}f(x)dx=n{\int }_0^{a}f(x)dx$

B. Wallis’s Theorem

Wallis’s Theorem provides a shortcut for evaluating definite integrals of powers of $\sin x$ or $\cos x$ from $0$ to $\pi /2$: ${\int }_0^{\pi /2}{\sin }^{n}xdx={\int }_0^{\pi /2}{\cos }^{n}xdx$

• When $n$ is even: The result involves factors of $\frac{n-1}{n}$, $\frac{n-3}{n-2}$, $\frac{n-5}{n-4},\dots ,\frac{3}{4},\frac{1}{2},\frac{\pi }{2}$.
• When $n$ is odd: The result involves factors of $\frac{n-1}{n}$, $\frac{n-3}{n-2}$, $\frac{n-5}{n-4},\dots ,\frac{4}{5},\frac{2}{3},1$.

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III. Reduction Formulas

Reduction formulas allow an integral of power $n$ ($I_n$) to be expressed in terms of an integral of a lower power ($I_{n-2}$).

1. For $\int {\sin }^{n}xdx$ ($I_n$): $I_n=\frac{-{\sin }^{n-1}x\cos x}{n}+\frac{n-1}{n}{I}_{n-2}$ (The source also states ${\int }_0^{\pi /2}{\sin }^{n}xdx=\frac{n-1}{n}{I}_{n-2}$).

2. For $\int {\tan }^{n}xdx$ ($I_n$): $I_n=\frac{{\tan }^{n-1}x}{n-1}-I_{n-2}$

3. For $\int e^{ax}{\cos }^{n}xdx$ ($I_n$): The reduction formula is derived by integration by parts multiple times, resulting in $I_n$ being defined recursively in terms of $I_{n-2}$: $$ I_n = \frac{a^2}{a^2 + n^2} \left{ (\cdots) + \frac{n(n-1)}{a^2} I_{n-2} \right}

(Note: The first part of the expression, denoted as $(\cdots)$ or $(1)$ in the source, is the terms derived from the integration by parts boundary terms). 4. **For $\int \sin^m x \cos^n x dx$ ($I_{m, n}$):** The source provides a relationship reducing $n$ to $n-2$: $$I_{m, n} = \frac{n-1}{m+n} I_{m, n-2} + \frac{\cos^{n-1} x \sin^{m+1} x}{m+n}$$ --- ### IV. Beta and Gamma Functions ^{These are defined as Eulerian integrals and are crucial for solving specific definite integrals. Both functions are defined only for positive values of their parameters ($m>0, n>0$).} #### A. Beta Function ($\text{B}(m, n)$) The Beta function is known as the **First Eulerian integral**. **Definition:** $$\text{B}(m, n) = \int_{0}^{1} x^{m-1} (1-x)^{n-1} dx$$ **Alternative Forms:** 1. $\text{B}(m, n) = \int_{0}^{\infty} \frac{y^{m-1}}{(1+y)^{m+n}} dy$ 2. $\text{B}(m, n) = \int_{0}^{\infty} \frac{y^{n-1}}{(1+y)^{m+n}} dy$ 3. $\text{B}(m, n) = 2 \int_{0}^{\pi/2} \sin^{2m-1} \theta \cos^{2n-1} \theta d\theta$ #### B. Gamma Function ($\Gamma(n)$) The Gamma function is known as the **Second Eulerian integral**. **Definition:** $$\Gamma(n) = \int_{0}^{\infty} e^{-x} x^{n-1} dx$$ **Properties of Gamma Function:** 1. $\Gamma(n+1) = n!$ (If $n$ is a positive integer) 2. $\Gamma(n+1) = n \Gamma(n)$ 3. $\Gamma(1) = 1$ 4. $\Gamma(n) \Gamma(1-n) = \frac{\pi}{\sin(n\pi)}$ (Note: The term for $\pi$ is partially obscured in the source but is implied by the structure $\frac{\pi}{\sin(\cdots)}$). #### C. Relation between Beta and Gamma Functions $$\text{B}(m, n) = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)}$$ **Key Formula Derived:** $$\int_{0}^{\pi/2} \sin^p \theta \cos^q \theta d\theta = \frac{\Gamma\left(\frac{p+1}{2}\right) \Gamma\left(\frac{q+1}{2}\right)}{2 \Gamma\left(\frac{p+q+2}{2}\right)}$$ --- ### V. Applications of Integration Integration is used to calculate geometric properties of curves, including Area, Length, and Volume/Surface Area of Revolution. #### A. Area (Rectangular Coordinates) - **Area bounded by $y=f(x)$ and the $x$-axis** between ordinates $x=a$ and $x=b$: $$\text{Area} = \int_{a}^{b} f(x) dx \quad \text{or} \quad \int_{a}^{b} y dx$$ - **Area bounded by $x=f(y)$ and the $y$-axis** between $y=a$ and $y=b$: $$\text{Area} = \int_{a}^{b} f(y) dy$$ - **Symmetry Tip:** A curve is symmetric along the $x$-axis if the function of $y$ has an even power (i.e., $y$ occurs as $y^2, y^4$, etc.). #### B. Length (Arc Length of Curves) The formula for arc length depends on the form of the curve definition: |Curve Form|Arc Length Formula ($L$) (from $a$ to $b$)| |:--|:--| |**Explicit ($y=f(x)$)**|$$L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$$| |**Explicit ($x=f(y)$)**|$$L = \int_{a}^{b} \sqrt{1 + \left(\frac{dx}{dy}\right)^2} dy$$| |**Parametric ($x=f(t), y=\Phi(t)$)**|$$L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$$| |**Polar ($r=f(\theta)$)**|$$L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta$$| #### C. Volume of Revolution This table summarizes the formulas used to calculate the volume of a solid generated by revolving a curve defined by $r=f(\theta)$, $y=f(x)$, or $x=f(y)$ around a specific axis. The integral must be evaluated between the defined limits, $x=a$ to $x=b$, $y=a$ to $y=b$, or $\theta=\alpha$ to $\theta=\beta$. | Axis of Revolution | Curve Form | Volume Formula ($V$) | | :------------------------------------ | :--------------------- | :------------------------------------------------------------------- | | **$x$-axis** | $y=f(x)$ (Rectangular) | $$V = \int_{a}^{b} \pi y^2 dx$$ | | **$y$-axis** | $x=f(y)$ (Rectangular) | $V = \int_{a}^{b} \pi x^2 dy$ | | **Initial line ($OX$ or $\theta=0$)** | $r=f(\theta)$ (Polar) | $$V = \int_{\alpha}^{\beta} \frac{2\pi}{3} r^3 \sin \theta d\theta$$ | | **Axis $OY$ or $\theta=\pi/2$** | $r=f(\theta)$ (Polar) | $V = \int_{\alpha}^{\beta} \frac{2\pi}{3} r^3 \cos \theta d\theta$ | #### D. Surface Area of Revolution | Axis of Revolution | Curve Form | Surface Area Formula ($S$) | Relevant Arc Length Definitions ($ds$) | | :---------------------------- | :--------------------- | :------------------------------------------------------------ | :------------------------------------------------------------------------------------------------------------------------------------------------ | | **$x$-axis** | $y=f(x)$ or Parametric | $S = \int_{a}^{b} 2\pi y ds$ | $\frac{ds}{dx} = \sqrt{1 + \left(\frac{dy}{dx}\right)^2}$ or $\frac{ds}{dt} = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}$ | | **$y$-axis** | $x=f(y)$ | $S = \int_{a}^{b} 2\pi x ds$ | $ds = \sqrt{1 + \left(\frac{dx}{dy}\right)^2} dy$ (implied by context) | | **Initial line ($\theta=0$)** | $r=f(\theta)$ (Polar) | $S = \int_{\alpha}^{\beta} 2\pi y \frac{ds}{d\theta} d\theta$ | $y = r \sin \theta$ and $\frac{ds}{d\theta} = \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}$ | **Definition of $ds$:** - $ds = \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$ - $ds = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$ (for parametric $x=f(t), y=\Phi(t)$) - $\frac{ds}{d\theta} = \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}$ (for polar $r=f(\theta)$) ---