Integral Calculus - Integration with log properties

  • In this section, we will learn about integration with logarithmic properties.
  • These properties help simplify the integration process.
  • Let’s start by reviewing some basic logarithmic properties.

Logarithmic Properties

  • The logarithmic properties are:
    • Product Rule:
      • ( \log(ab) = \log(a) + \log(b) )
    • Quotient Rule:
      • ( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) )
    • Power Rule:
      • ( \log(a^n) = n\log(a) )

Integration of Natural Log Functions

  • The natural logarithm function is denoted by ( \ln(x) ).
  • To integrate the natural log function, we can use the following formula: [ \int \ln(x) , dx = x(\ln(x) - 1) + C ]
  • Example:
    • Find (\int \ln(x) , dx) [ \int \ln(x) , dx = x(\ln(x) - 1) + C ]

Integration of Other Log Functions

  • Apart from the natural logarithm, we can also integrate other log functions.
  • The general formula for the integration of ( \log_a(x) ) is: [ \int \log_a(x) , dx = \frac{x(\log_a(x) - 1)}{\ln(a)} + C ]
  • Example:
    • Evaluate (\int \log_3(x) , dx) [ \int \log_3(x) , dx = \frac{x(\log_3(x) - 1)}{\ln(3)} + C ]

Integration involving Logarithmic Functions

  • Sometimes, we come across integration problems that involve both logarithmic and other functions.
  • To solve such integration problems, we need to use appropriate techniques like substitution or integration by parts.
  • Example:
    • Find (\int x \ln(x) , dx)

Integration of Exponential Functions

  • Exponential functions are the inverse of logarithmic functions.
  • To integrate exponential functions, we can use the property: [ \int e^x , dx = e^x + C ]
  • Example:
    • Evaluate (\int e^{2x} , dx) [ \int e^{2x} , dx = \frac{1}{2} e^{2x} + C ]

Integration involving Log and Exponential Functions

  • Integration problems involving both log and exponential functions can be solved using substitution or integration by parts.
  • These problems may require some algebraic manipulation before integrating.
  • Example:
    • Find (\int e^x \ln(x) , dx)

Integration of Logarithmic Expressions

  • Sometimes, we may encounter integration problems with expressions involving logarithms.
  • These problems can be solved by applying logarithmic properties and appropriate integration techniques.
  • Example:
    • Evaluate (\int \frac{\ln(x)}{x} , dx)

Integration of Logarithmic Functions with Constants

  • When integrating logarithmic functions with constants, we need to consider the coefficient of the log function.
  • The general formula for integration in such cases is: [ \int a \log(x) , dx = a(x \log(x) - x) + C ]
  • Example:
    • Determine (\int 3 \ln(x) , dx) [ \int 3 \ln(x) , dx = 3(x \ln(x) - x) + C ]

Integration of Log Functions with Power Rule

  • Logarithmic functions raised to a power can be integrated using the power rule.
  • To integrate ( \left(\log(x)\right)^n ), we can apply the following formula: [ \int \left(\log(x)\right)^n , dx = \frac{x \left(\log(x)\right)^{n+1}}{n+1} - \frac{x \left(\log(x)\right)^n}{n(n+1)} + C ]
  • Example:
    • Find (\int \left(\log(x)\right)^2 , dx) [ \int \left(\log(x)\right)^2 , dx = \frac{x \left(\log(x)\right)^3}{3} - \frac{x \left(\log(x)\right)^2}{2\cdot3} + C ] '

Properties of Definite Integrals

  • The definite integral has several properties that we can make use of:
    • Linearity Property:
      • (\int_a^b kf(x) , dx = k\int_a^b f(x) , dx) for a constant (k)
    • Summation Property:
      • (\int_a^b [f(x) + g(x)] , dx = \int_a^b f(x) , dx + \int_a^b g(x) , dx)
    • Constant Property:
      • (\int_a^a f(x) , dx = 0)

Integration by Substitution

  • Integration by substitution is a method that allows us to solve certain types of integrals.
  • The process involves substituting part of the integrand with a new variable.
  • Steps for integration by substitution:
    1. Identify a suitable substitution (usually a function and its derivative).
    2. Substitute the function and its derivative into the integral.
    3. Evaluate the new integral.
    4. Substitute the original variable back in.

Example: Integration by Substitution

  • Let’s solve the integral (\int 2x \cos(x^2) , dx) using integration by substitution.
  • Step 1: Let (u = x^2), then (\frac{du}{dx} = 2x) (Derivative of (u)).
  • Step 2: Substitute (x^2) for (u) and (2x) for (\frac{du}{dx}) in the integral.
  • Step 3: The new integral becomes (\int \cos(u) , du).
  • Step 4: Integrating (\cos(u)) gives (\sin(u) + C).
  • Step 5: Substitute (u) back to (x^2) in the result.

Integration by Parts

  • Integration by parts is another technique for evaluating integrals.
  • It is based on the product rule for differentiation.
  • The formula for integration by parts is: [ \int u , dv = uv - \int v , du ]
  • Steps for integration by parts:
    1. Choose (u) and (dv) from the original integrand.
    2. Calculate (du) and (v) using the derivatives and antiderivatives.
    3. Apply the formula (\int u , dv = uv - \int v , du).
    4. Simplify and solve the resulting integrals.

Example: Integration by Parts

  • Let’s solve the integral (\int x \cos(x) , dx) using integration by parts.
  • Step 1: Choose (u = x) and (dv = \cos(x) , dx).
  • Step 2: Calculate (du) and (v) by differentiating and integrating the chosen functions.
  • Step 3: Apply the formula (\int u , dv = uv - \int v , du).
  • Step 4: Simplify and solve the resulting integral.
  • Step 5: Do any necessary algebra to obtain the final answer.

Trigonometric Substitutions

  • Trigonometric substitutions are used to solve integrals that involve radicals and trigonometric functions.
  • The substitutions depend on the form of the integral and the trigonometric identity being used.
  • The three common substitutions are:
    1. If the integral involves (\sqrt{a^2 - x^2}), use (x = a \sin(\theta)).
    2. If the integral involves (\sqrt{a^2 + x^2}), use (x = a \tan(\theta)).
    3. If the integral involves (\sqrt{x^2 - a^2}), use (x = a \sec(\theta)).

Example: Trigonometric Substitution

  • Let’s solve the integral (\int \frac{1}{\sqrt{9 - x^2}} , dx) using a trigonometric substitution.
  • Step 1: Use the substitution (x = 3 \sin(\theta)).
  • Step 2: Calculate (dx) and substitute into the integral.
  • Step 3: Substitute (\sqrt{9 - x^2}) with (3 \cos(\theta)).
  • Step 4: Simplify and solve the integral using trigonometric identities.
  • Step 5: Do any necessary algebra to obtain the final answer.

Integration of Rational Functions

  • Rational functions are functions that are represented as the quotient of two polynomials.
  • To integrate rational functions, we use a method called partial fraction decomposition.
  • Steps for integrating rational functions with partial fraction decomposition:
    1. Factor the denominator into irreducible factors.
    2. Write the rational function as a sum of partial fractions.
    3. Determine the unknown coefficients by equating the numerators.
    4. Integrate each partial fraction separately.
    5. Combine the integrals to obtain the final answer.

Example: Integration of Rational Functions

  • Let’s solve the integral (\int \frac{3x - 1}{(x - 2)(x + 1)} , dx) using partial fraction decomposition.
  • Step 1: Factor the denominator into ((x - 2)(x + 1)).
  • Step 2: Write the rational function as (\frac{A}{x - 2} + \frac{B}{x + 1}).
  • Step 3: Equate the numerators to determine the unknown coefficients (A) and (B).
  • Step 4: Integrate each partial fraction separately.
  • Step 5: Combine the integrals to obtain the final answer.

Integration of Trigonometric Functions

  • Trigonometric functions can be integrated using various techniques.
  • The common trigonometric integrals are:
    • (\int \sin(x) , dx = -\cos(x) + C)
    • (\int \cos(x) , dx = \sin(x) + C)
    • (\int \tan(x) , dx = -\ln|\cos(x)| + C)
    • (\int \sec(x) , dx = \ln|\sec(x) + \tan(x)| + C)
    • (\int \csc(x) , dx = -\ln|\csc(x) + \cot(x)| + C) '

Slide 21

  • Solving Integration Problems involving Trigonometric Substitutions
  • Let’s look at an example:
    • Evaluate (\int \sqrt{1 - x^2} , dx)
  • Step 1: Use the substitution (x = \sin(\theta))
  • Step 2: Calculate (dx) and substitute into the integral.
  • Step 3: Substitute (\sqrt{1 - x^2}) with (\cos(\theta)).
  • Step 4: Simplify and solve the integral using trigonometric identities.
  • Step 5: Do any necessary algebra to obtain the final answer.

Slide 22

  • Solving Integration Problems involving Trigonometric Substitutions (Contd.)
  • Example:
    • Evaluate (\int \frac{1}{x^2 \sqrt{x^2 - 9}} , dx)
  • Step 1: Use the substitution (x = 3 \sec(\theta))
  • Step 2: Calculate (dx) and substitute into the integral.
  • Step 3: Substitute (\sqrt{x^2 - 9}) with (3 \tan(\theta)).
  • Step 4: Simplify and solve the integral using trigonometric identities.
  • Step 5: Do any necessary algebra to obtain the final answer.

Slide 23

  • Solving Integration Problems involving Rational Functions
  • Rational functions are functions that can be expressed as the ratio of two polynomials.
  • To integrate rational functions, we can use the method of partial fractions.
  • Step 1: Factor the denominator into irreducible factors.
  • Step 2: Write the rational function as a sum of partial fractions.
  • Step 3: Determine the unknown coefficients by equating the numerators.
  • Step 4: Integrate each partial fraction separately.
  • Step 5: Combine the integrals to obtain the final answer.

Slide 24

  • Solving Integration Problems involving Rational Functions (Contd.)
  • Example:
    • Evaluate (\int \frac{4x - 1}{x^2 + 4x + 4} , dx)
  • Step 1: Factor the denominator into ((x + 2)(x + 2))
  • Step 2: Write the rational function as (\frac{A}{x + 2} + \frac{B}{(x + 2)^2})
  • Step 3: Equate the numerators to determine the unknown coefficients (A) and (B)
  • Step 4: Integrate each partial fraction separately.
  • Step 5: Combine the integrals to obtain the final answer.

Slide 25

  • Solving Integration Problems involving Logarithmic Functions
  • Logarithmic functions can be integrated using various techniques and properties.
  • Some common techniques include substitution and integration by parts.
  • It is important to simplify the expression before integrating.
  • Example:
    • Evaluate (\int x \ln(x + 1) , dx)
  • We can solve this using integration by parts.
  • Choose (u = \ln(x + 1)) and (dv = x , dx), then apply the formula: (\int u , dv = uv - \int v , du)

Slide 26

  • Solving Integration Problems involving Logarithmic Functions (Contd.)
  • Example:
    • Evaluate (\int \frac{\ln(x)}{x^2} , dx)
  • We can solve this using substitution.
  • Choose (u = \ln(x)), then calculate (du).
  • Substitute (u) and (du) in the integral and solve.

Slide 27

  • Solving Integration Problems involving Logarithmic Expressions
  • Logarithmic expressions can be integrated by applying logarithmic properties.
  • Here’s an example:
    • Evaluate (\int \ln(x^2) , dx)
  • We can rewrite (\ln(x^2)) as (2 \ln(x)) using the logarithmic properties.
  • Solve the integral (\int 2 \ln(x) , dx) using the appropriate technique.

Slide 28

  • Solving Integration Problems involving Logarithmic Expressions (Contd.)
  • Example:
    • Evaluate (\int \frac{\ln(x)}{x} , dx)
  • We can solve this using substitution.
  • Choose (u = \ln(x)), then calculate (du).
  • Substitute (u) and (du) in the integral and solve.

Slide 29

  • Summary
  • In this lecture, we covered integration with logarithmic properties.
  • We learned how to integrate natural logarithm functions and other log functions.
  • We also studied integration involving logarithmic and exponential functions.
  • Furthermore, we discussed integration of logarithmic expressions and functions with power rules.
  • Lastly, we explored properties of definite integrals and the techniques of substitution, integration by parts, trigonometric substitutions, and integration of rational functions.

Slide 30

  • Thank You
  • Any Questions?