Integration Techniques

Table of Contents

1. Common Functions

1.1. Trigonometric Functions

  • Half angle formula if the power is even for sine and cosine.
  • Substitution for odd power of sine and cosine.
  • Sum formula if the integrand is the multiplication of single powered sine and cosine.
  • Substitution for even power of tangent and secent.
  • List of integrals of trigonometric functions - Wikipedia

1.2. Rational Functions

1.2.1. Polynomial Division

1.2.2. Partial Fraction Decomposition

1.2.2.1. Heaviside Cover-Up Method

For a rational function of the form:

\begin{equation*} R(x) = \frac{f(x)}{g(x)} \end{equation*}

where \( f(x), g(x) \) is a polynomial, with the set of distinct roots of \( g(x) \) \( \{ \alpha_i\}_i \) and the multiplicity of each being \( \{m_i\}_i \), can always be decomposed into partial fraction:

\begin{equation*} R(x) = \sum_{i} \sum_{j=1}^{m_i}\frac{A_{ij}}{(x-\alpha_i)^j}. \end{equation*}

where \( A_{ij} \) being real numbers.

\( A_{im_i} \) can be calculated by removing the factors including the \( i \)th root from the denominator:

\begin{equation*} A_{im_i} = \lim_{x\to \alpha_i} (x-\alpha_i)^{m_i}R(x). \end{equation*}

Further, in the case of repeated roots, one way to calculate \( A_{ij} \) is taking derivative:

\begin{equation*} A_{ij} = \lim_{x\to \alpha_i}\left( \frac{d}{dx}\right)^{m_i-j} (x-\alpha_i)^{m_i}R(x). \end{equation*}

\( A_{ij} \) can be also be found with expanding the partial fraction and setting \( x \) to any value.

1.2.3. Trigonometric Substitution

1.2.4. Common Antiderivatives

1.2.4.1. Inverse Trigonometric Functions
1.2.4.1.1. Composition
  • \(\sin(\cos^{-1} x) = \sqrt{1-x^2}\)
  • \(\sec(\tan^{-1}x) = \sqrt{1+x^2}\)
1.2.4.1.2. Derivatives
  • The denominator is the derivative of the normal function in terms of the normal function as \(x\).
  • \[ \frac{d}{dx}\sin^{-1}x = \frac{1}{\sqrt{1-x^2}} \]
  • \[ \frac{d}{dx}\tan^{-1}x = \frac{1}{1+x^2} \]
  • \[ \frac{d}{dx}\sec^{-1}x = \frac{1}{|x|\sqrt{x^2-1}} \]
1.2.4.2. Inverse Hyperbolic Functions
1.2.4.2.1. Equivalent Expression
  • \(\sinh^{-1}x = \ln\left( x + \sqrt{x^2+1}\right)\)
  • \(\cosh^{-1}x = \ln\left(x+\sqrt{x^2-1}\right)\)
    • \[ \begin{align*} \cosh^{-1} &y = x \\ \implies &y+\sqrt{y^2 - 1} = e^x\\ \implies &y^2 - 1 = e^{2x} - 2ye^x + y^2 \\ \implies & y = \frac{e^{2x}+1}{2e^x} = \cosh x \end{align*} \]
  • \( \displaystyle \tanh^{-1}x = \frac{1}{2}\ln\frac{1+x}{1-x} \)
1.2.4.2.2. Derivatives
  • \[ \frac{d}{dx}\sinh^{-1}x=\frac{1}{\sqrt{1+x^2}} \]
  • \[ \frac{d}{dx}\cosh^{-1}x = \frac{1}{\sqrt{x^2-1}} \]
  • \[ \frac{d}{dx}\tanh^{-1}x = \frac{1}{1-x^2} \]

2. Integration by Substitution

  • \(u\)-Substitution, Reverse Chain Rule, Change of Variables

2.1. Common Substitution

  • \(e^x \rightsquigarrow u\)
    • Then \(dx = du/u\). See if this would simplify.
  • \(\sqrt{ax^n + b} \rightsquigarrow u\)

2.2. Trigonometric Substitution

2.3. Hyperbolic Substitution

2.4. Tangent Half-Angle Substitution

  • Universal Trigonometric Substitution, Weierstrass Substitution
  • \[ \tan \left( \frac{x}{2} \right) \leadsto u \]

  • This transforms trigonometric functions as

    \begin{align} \sin x &= \frac{2u}{u^{2}+1} \\ \cos x & = \frac{u^{2}-1}{u^{2}+1} \\ \tan x & = \frac{2u}{u^{2}-1} \\ dx & = \frac{2}{u^{2}+1}du \end{align}

2.5. Euler Substitution

  • Method for evaluating integrals of the form: \[ \int R(x, \sqrt{ax^2 + bx + c})\,dx \] where \(R\) is a rational function.

2.5.1. Euler's First Substitution

  • When \(a > 0\).
  • It is favorable to have \(b = 0\).
  • \[ \sqrt{ax^2+bx+c} = \pm x\sqrt{a} + t. \]
  • \[ x = \frac{t^2-c}{b\mp2t\sqrt{a}} \]
2.5.1.1. Examples
  • \[ \int \sqrt{1+x^2}\,dx \]
    • with:
    • \[ t = x + \sqrt{x^2 + 1} \]
    • \[ x = \frac{t^2 - 1}{2t} = \frac{t}{2} - \frac{1}{2t} \]
    • \[ \sqrt{x^2 + 1} = \frac{t^2 + 1}{2t} = \frac{t}{2} + \frac{1}{2t} \]
    • \[ dx = \frac{t^2 + 1}{2t^2}\,dt = \left(\frac{1}{2} + \frac{1}{2t^2}\right)\,dt \]
  • \[ \int\frac{dx}{\sqrt{x^2 + c}} \]
    • with:
    • \[ t = x+\sqrt{x^2+c} \]
    • \[ x = \frac{t^2 - c}{2t} \]
    • \[ \sqrt{x^2+c} = \frac{t^2 + c}{2t} \]
    • \[ dx = \frac{t^2 + c}{2t^2}\,dt \]
  • \[ \int\frac{dx}{x\sqrt{x^2 + 4x - 4}} \]
    • with:
    • \[ t = \sqrt{x^2 + 4x -4} - x \]
    • \[ x = \frac{t^2 + 4}{4 - 2t} \]

2.5.2. Euler's Second Substitution

  • When \(c > 0\).
    • Two \(x\) terms on the denominator.
    • \[ \sqrt{ax^2 + bx + c} = xt \pm \sqrt{c}. \]
    • \[ x = \frac{b \mp 2t\sqrt{c}}{t^2 - a} \]
2.5.2.1. Examples
  • \[ \int\frac{dx}{x\sqrt{-x^2 + x + 2}} \]
  • with:
  • \[ \sqrt{-x^2 + x+ 2} = xt + \sqrt{2} \]
  • \[ x = \frac{1-2\sqrt{2}t}{t^2 + 1} \]
  • \[ \sqrt{-x^2 + x + 2} = \frac{-\sqrt{2}t^2 + t + \sqrt{2}}{t^2 + 1} \]
  • \[ dx = \frac{2\sqrt{2}t^2 - 2t - 2\sqrt{2}}{(t^2 + 1)^2}\,dt \]

2.5.3. Euler's Third Substitution

  • When the polynomial \(ax^2 + bx + c\) has real roots \(\alpha, \beta\).
  • \[ \sqrt{ax^2 + bx + c} = \sqrt{a(x-\alpha)(x-\beta)}= (x-\alpha)t. \]
  • \[ x = \frac{a\beta - \alpha t^2}{a - t^2} \]
2.5.3.1. Examples
  • \[ \int\frac{x^2\,dx}{\sqrt{-x^2 + 3x -2}} \]
    • with:
    • \[ \sqrt{-(x-2)(x-1)} = (x-2)t \]
    • \[ x = \frac{-2t^2 - 1}{-t^2 - 1} \]
    • \[ dx = \frac{2t}{(-t^2 - 1)^2}\,dt \]
    • \[ \sqrt{-x^2+3x - 2} = \frac{t}{-t^2 - 1} \]

3. Integration by Parts

3.1. Tabular Integration

  • DI Method
  • \[ \int f(x)g(x) \,dx \]
  • Integrate a product by calculating the table:
  D I
\(+\) \(f(x)\) \(g(x)\)
\(-\) \(f^{(1)}(x)\) \(g^{(-1)}(x)\)
\(+\) \(f^{(2)}(x)\) \(g^{(-2)}(x)\)
\(...\) \(...\) \(...\)
\((-1)^n\) \(0\) \(g^{(-n)}(x)\)
  • The antiderivative is obtained by the sum of the products of diagonals:
    • \[ \sum_{k=0}^{n-1} (-1)^kf^{(k)}(x)g^{(-k-1)}(x) + \int (-1)^n f^{(n)}(x)g^{(-n)}\,dx \]
  • The process can be stopped at any stage, leaving an unresolved integral.

4. Integration by Reduction Formulae

  • \[ \int \sec^n ax\, dx = \frac{1}{a}\tan ax\sec^{n-2} ax + (n-2)\int\sec^{n-2} ax\, dx \]

5. Integrand Transformation

  • Very obscure. Just make it integrable.
  • Multiply by 1, Add 0.

5.1. Matching the Denominator

  • Transform the numerator such that: \[ \int\frac{g(x)}{f(x)}\,dx \rightsquigarrow \int\frac{Af(x) + Bf'(x)}{f(x)}\,dx \] where \(A, B\) are constants.
  • Consider this if the numenator is part of the denominator or the derivative of it.

5.1.1. Examples

  • \[ \int \frac{dx}{1+e^x} \]
  • Add 0: \[ = \int \frac{1+e^x}{1+e^x} - \frac{e^x}{1+e^x}\,dx = x - \ln(1+e^x) + C. \]

5.2. Completing the Factor

  • Multiply by the factor that simplifies on both the numerator and the denominator.
  • Multiply 1: \[ t^2 - t + 1 = \frac{t^3 + 1}{t+1}. \]

5.3. Expanding into Power Series

5.4. Introducing Integrals or Derivatives

  • Substitute part of the integrand as the integral or derivative of another function, hopefully simpler to calculate.
  • The integral may be the zeroth integral, that is, the evaluation at the limits.
  • Then use the interchange of integrals or integration by parts.

6. Differentiation Under the Integral Sign

  • Feynman Technique
  • Differentiate with respect to some variable that is not being integrated.
  • The variable may as well be created.

7. Ansatz

  • Make an educated guess of the antiderivative.
  • The antiderivative would most likely contain the antiderivatives of the parts of the integrand.

8. Definite Integral

8.1. Reciprocation

\( x \rightsquigarrow 1/u \)

\[ \int_a^{1/a}f(x)\,dx = \int_a^{1/a} f \left( \frac{1}{x} \right) \frac{1}{x^2} \,dx \]

It include the case with the bound \(0\) to \(\infty\).

Consider it, especially when \( 1 + x^2 \) is in the denominator.

8.2. King's Rule

Border Flip substitution \[ \int_{a}^{b} f(x) \, dx =\int_{a}^{b} f(a+b-x) \, dx \]

Especially, \[ \int_{-a}^a f(x)\,dx = \int_{-a}^a f(-x)\,dx. \]

8.3. Glasser's Master Theorem

  • For real numbers \(a, \{a_i\}, \{b_i\}\), the substitution \[ u = x - a \sum_{n=1}^N\frac{|a_n|}{x-b_n} \] does not change the result: \[ \mathcal{P}\int_{-\infty}^\infty f(u)\,dx = \mathcal{P}\int_{-\infty}^\infty f(x)\,dx, \] where \(\mathcal{P}\) denote the Cauchy principal value.

8.4. Symmetry

  • If \( f(a+b - x) = f(x) \), then

\[ \int_a^bf(x)\,dx = 2\int_a^{(a+b)/2}f(x)\,dx. \]

8.5. Normalization

  • \[ \int_0^{x_0} R\left(\frac{f(x)}{f(x_0)}\right)\,dx \rightsquigarrow \int_0^c \tilde{R}(f(t))\,dt. \]

8.6. Averaging after Substitution

  • Used when the bound remains unchanged during substitution

\[ I = \int_a^bf(x)\,dx = \frac{1}{2} \int_a^b(f(x) + \tilde{f}(x))\,dx. \]

9. Composition of Techniques

9.1. Inverse Function

  • Integration of inverse function
  • \[ \int f^{-1}(x)\,dx \]
    • Substitution: \(f^{-1}(x) = u\), \(x = f(u)\), \(dx = f'(u)\,du\).

    • Integration by part:

      \begin{align*} \int uf'(u)\,du &= uf(u) - \int f(u)\,du \\ &= uf(u) - F(u) + C\\ &= xf^{-1}(x) - F(f^{-1}(x)) + C. \end{align*}
    • Note that this is calculating the area of the rectangle \(xf^{-1}(x)\), and taking away the usual integral \(F(f^{-1}(x))\).

9.1.1. Examples

  • \[ \int \sin^{-1}(x)\,dx = x\sin^{-1}(x) + \sqrt{1-x^2} + C \]
  • \[ \int \cos^{-1}(x)\,dx = x\cos^{-1}(x) - \sqrt{1-x^2} + C \]
  • \[ \int \tan^{-1}(x)\,dx = x\tan^{-1}(x) - \frac{1}{2}\ln(1+x^2) + C \]
  • \[ \int \ln x\,dx = x\ln x - x + C \]
  • This technique is useful in the calculation of the inverse cumulative distribution, useful for generating samples from a probability distribution.

10. Others

11. References

Footnotes:

1

WHY are we finding pi HERE? - YouTube

Created: 2025-05-06 Tue 23:34