Indefinite Integral - AREA FUNCTION

  • The indefinite integral is a reverse process of differentiation.
  • It finds the antiderivative or the original function from the derivative of a given function.
  • Mathematically, if F'(x) = f(x), then F(x) is an antiderivative of f(x). Example:
  • Find the antiderivative of f(x) = 2x + 3. Solution:
  • F(x) = ∫ (2x + 3) dx
  • F(x) = ∫ 2x dx + ∫ 3 dx
  • F(x) = x^2 + 3x + C, where C is the constant of integration.

Evaluation of Definite Integral

  • The definite integral is used to calculate the area under a curve between two points.
  • It gives a numerical value as the result.
  • Mathematically, if F'(x) = f(x), then the definite integral of f(x) from a to b is denoted as ∫ [a, b] f(x) dx. Example:
  • Evaluate the definite integral ∫ [1, 3] (4x^2 + 3x) dx. Solution:
  • ∫ [1, 3] (4x^2 + 3x) dx = [ (x^3) + (3/2)x^2 ] [1, 3]
  • Plug in the values: (3^3 + (3/2)(3^2)) - (1^3 + (3/2)(1^2))
  • Simplifying, we get 54 - 4 = 50.

Properties of Indefinite Integrals

  • Linearity: ∫ (af(x) + bg(x)) dx = a∫ f(x) dx + b∫ g(x) dx
  • Constant Multiple: ∫ af(x) dx = a∫ f(x) dx (where a is a constant)
  • Sum/Difference Rule: ∫ (f(x) ± g(x)) dx = ∫ f(x) dx ± ∫ g(x) dx
  • Power Rule: ∫ x^n dx = (1/(n+1)) * x^(n+1) + C (where n is any real number) Example:
  • Evaluate ∫ (3x^2 + 5x - 2) dx. Solution:
  • ∫ (3x^2 + 5x - 2) dx = (3/3) * x^3 + (5/2) * x^2 - 2x + C
  • Simplifying, we get x^3 + (5/2) * x^2 - 2x + C.

Integration by Substitution

  • Integration by substitution is used to simplify integrals by substituting variables.
  • It requires expressing a function in terms of another variable.
  • The integral of f(g(x))*g'(x) can be transformed by substituting u = g(x). Example:
  • Evaluate ∫ 2x sin(x^2) dx. Solution:
  • Let u = x^2.
  • Differentiating, we get du = 2x dx.
  • Substituting in the integral, we have ∫ sin(u) du = -cos(u) + C.
  • Simplifying, we get -cos(x^2) + C.

Integration by Parts

  • Integration by parts is used to evaluate the integral of a product of two functions.
  • It involves splitting the integral into two parts and applying the product rule of differentiation.
  • The formula for integration by parts is ∫ u dv = uv - ∫ v du. Example:
  • Evaluate ∫ x cos(x) dx. Solution:
  • Let u = x and dv = cos(x) dx.
  • Differentiating u, we get du = dx.
  • Integrating dv, we get v = sin(x).
  • Applying the formula, we have ∫ x cos(x) dx = x sin(x) - ∫ sin(x) dx.
  • Evaluating the integral, we get x sin(x) + cos(x) + C.

Integration of Trigonometric Functions

  • The integrals of trigonometric functions can be evaluated using standard identities.
  • Some common integrals include:
    • ∫ sin(x) dx = -cos(x) + C
    • ∫ cos(x) dx = sin(x) + C
    • ∫ sec^2(x) dx = tan(x) + C
    • ∫ csc^2(x) dx = -cot(x) + C
    • ∫ sec(x) tan(x) dx = sec(x) + C
    • ∫ csc(x) cot(x) dx = -csc(x) + C Example:
  • Evaluate ∫ sin^3(x) cos^2(x) dx. Solution:
  • Using the identity sin^2(x) = 1 - cos^2(x), we can rewrite the integral as ∫ sin(x) (1 - cos^2(x)) cos^2(x) dx.
  • Expanding and simplifying, we get ∫ sin(x) cos^2(x) dx - ∫ sin(x) cos^4(x) dx.
  • Applying the known integrals, we have -(1/3) cos^3(x) + (1/5) cos^5(x) + C.

Integration of Exponential and Logarithmic Functions

  • The integrals of exponential and logarithmic functions can be evaluated using specific rules.
  • Some common integrals include:
    • ∫ e^x dx = e^x + C
    • ∫ a^x dx = (a^x) / ln(a) + C
    • ∫ log(a, x) dx = x (log(a, x) - 1) + C
    • ∫ ln(x) dx = x ln(x) - x + C Example:
  • Evaluate ∫ e^x ln(e^x) dx. Solution:
  • Using the property ln(a^x) = x ln(a), we can rewrite the integral as ∫ x e^x dx.
  • Applying the known integral ∫ x e^x dx = x e^x - e^x + C.

Integration of Rational Functions

  • Rational functions are the ratios of two polynomials.
  • Integrating rational functions involves partial fraction decomposition.
  • The steps for integrating rational functions include:
    1. Factorizing the denominator.
    2. Setting up the partial fraction decomposition.
    3. Finding the constants.
    4. Integrating each term separately. Example:
  • Evaluate ∫ (x^2 + 2x + 1) / (x^3 + 3x^2 + 3x) dx. Solution:
  • Factorizing the denominator, we get (x)(x+1)^2.
  • Setting up the partial fraction decomposition, we have A/(x) + B/(x+1) + C/(x+1)^2.
  • Finding the constants, we find A = -1, B = 3, and C = -2.
  • Integrating each term separately, we get -ln|x| + 3ln|x+1| - 2/(x+1) + C.

Indefinite Integral - AREA FUNCTION

  • The indefinite integral is a reverse process of differentiation.
  • It finds the antiderivative or the original function from the derivative of a given function.
  • Mathematically, if F'(x) = f(x), then F(x) is an antiderivative of f(x}. Example:
  • Find the antiderivative of f(x) = 2x + 3. Solution:
  • F(x) = ∫ (2x + 3) dx
  • F(x) = ∫ 2x dx + ∫ 3 dx
  • F(x) = x^2 + 3x + C, where C is the constant of integration.

Evaluation of Definite Integral

  • The definite integral is used to calculate the area under a curve between two points.
  • It gives a numerical value as the result.
  • Mathematically, if F'(x) = f(x), then the definite integral of f(x) from a to b is denoted as ∫ [a, b] f(x) dx. Example:
  • Evaluate the definite integral ∫ [1, 3] (4x^2 + 3x) dx. Solution:
  • ∫ [1, 3] (4x^2 + 3x) dx = [ (x^3) + (3/2)x^2 ] [1, 3]
  • Plug in the values: (3^3 + (3/2)(3^2)) - (1^3 + (3/2)(1^2))
  • Simplifying, we get 54 - 4 = 50.

Properties of Indefinite Integrals

  • Linearity: ∫ (af(x) + bg(x)) dx = a∫ f(x) dx + b∫ g(x) dx
  • Constant Multiple: ∫ af(x) dx = a∫ f(x) dx (where a is a constant)
  • Sum/Difference Rule: ∫ (f(x) ± g(x)) dx = ∫ f(x) dx ± ∫ g(x) dx
  • Power Rule: ∫ x^n dx = (1/(n+1)) * x^(n+1) + C (where n is any real number) Example:
  • Evaluate ∫ (3x^2 + 5x - 2) dx. Solution:
  • ∫ (3x^2 + 5x - 2) dx = (3/3) * x^3 + (5/2) * x^2 - 2x + C
  • Simplifying, we get x^3 + (5/2) * x^2 - 2x + C.

Integration by Substitution

  • Integration by substitution is used to simplify integrals by substituting variables.
  • It requires expressing a function in terms of another variable.
  • The integral of f(g(x))*g'(x) can be transformed by substituting u = g(x). Example:
  • Evaluate ∫ 2x sin(x^2) dx. Solution:
  • Let u = x^2.
  • Differentiating, we get du = 2x dx.
  • Substituting in the integral, we have ∫ sin(u) du = -cos(u) + C.
  • Simplifying, we get -cos(x^2) + C.

Integration by Parts

  • Integration by parts is used to evaluate the integral of a product of two functions.
  • It involves splitting the integral into two parts and applying the product rule of differentiation.
  • The formula for integration by parts is ∫ u dv = uv - ∫ v du. Example:
  • Evaluate ∫ x cos(x) dx. Solution:
  • Let u = x and dv = cos(x) dx.
  • Differentiating u, we get du = dx.
  • Integrating dv, we get v = sin(x).
  • Applying the formula, we have ∫ x cos(x) dx = x sin(x) - ∫ sin(x) dx.
  • Evaluating the integral, we get x sin(x) + cos(x) + C.

Integration of Trigonometric Functions

  • The integrals of trigonometric functions can be evaluated using standard identities.
  • Some common integrals include:
    • ∫ sin(x) dx = -cos(x) + C
    • ∫ cos(x) dx = sin(x) + C
    • ∫ sec^2(x) dx = tan(x) + C
    • ∫ csc^2(x) dx = -cot(x) + C
    • ∫ sec(x) tan(x) dx = sec(x) + C
    • ∫ csc(x) cot(x) dx = -csc(x) + C Example:
  • Evaluate ∫ sin^3(x) cos^2(x) dx. Solution:
  • Using the identity sin^2(x) = 1 - cos^2(x), we can rewrite the integral as ∫ sin(x) (1 - cos^2(x)) cos^2(x) dx.
  • Expanding and simplifying, we get ∫ sin(x) cos^2(x) dx - ∫ sin(x) cos^4(x) dx.
  • Applying the known integrals, we have -(1/3) cos^3(x) + (1/5) cos^5(x) + C.

Integration of Exponential and Logarithmic Functions

  • The integrals of exponential and logarithmic functions can be evaluated using specific rules.

  • Some common integrals include:

    • ∫ e^x dx = e^x + C
    • ∫ a^x dx = (a^x) / ln(a) + C
    • ∫ log(a, x) dx = x (log(a, x) - 1) + C
    • ∫ ln(x) dx = x ln(x) - x + C Example:

  • Evaluate ∫ e^x ln(e^x) dx. Solution:

  • Using the property ln(a^x) = x ln(a), we can rewrite the integral as ∫ x e^x dx.

  • Applying the known integral ∫ x e^x dx = x e^x - e^x + C.

Integration Techniques

  • There are various techniques to evaluate integrals.
  • Some common techniques include:
    • Integration by parts
    • Integration by substitution
    • Trigonometric substitution
    • Partial fractions
    • Trigonometric identities
    • Logarithmic differentiation

Integration by Trigonometric Substitution

  • Trigonometric substitution is used to simplify integrals with radicals involving trigonometric functions.
  • The substitution involves using appropriate trigonometric identities.
  • Some common substitutions include:
    • x = a sin(t)
    • x = a cos(t)
    • x = a tan(t) Example:
  • Evaluate ∫ √(4 - x^2) dx. Solution:
  • Using the substitution x = 2 sin(t), we have dx = 2 cos(t) dt.
  • Substituting in the integral, we get ∫ √(4 - 4sin^2(t)) * 2cos(t) dt.
  • Simplifying, we get ∫ 2 cos^2(t) dt.
  • Applying the trigonometric identity cos^2(t) = (1 + cos(2t))/2, we have ∫ (1 + cos(2t)) dt.
  • Evaluating the integral, we get t + (sin(2t))/2 + C.
  • Substituting back t = sin^(-1)(x/2), we have the final answer as sin^(-1)(x/2) + (sin(x))/2 + C.

Integration of Algebraic Fractions

  • Algebraic fractions are ratios of polynomials.
  • Integrating algebraic fractions involves decomposing the fractions into partial fractions.
  • The steps include:
    1. Factorize the denominator.
    2. Express the fraction as the sum of partial fractions.
    3. Determine the unknown coefficients.
    4. Integrate each term separately. Example:
  • Evaluate ∫ (7x - 1) / (x^2 - x - 2) dx. Solution:
  • Factoring the denominator, we have (x - 2)(x + 1).
  • Expressing the fraction as partial fractions, we have ∫ [A / (x - 2)] + [B / (x + 1)] dx.
  • Determining the coefficients, we find A = -3 and B = 4.
  • Integrating each term separately, we get -3 ln|x - 2| + 4 ln|x + 1| + C.

Integration of Rational Functions with Quadratic Denominators

  • Integrating rational functions with quadratic denominators involves partial fraction decomposition.
  • The steps include:
    1. Factorize the denominator.
    2. Express the fraction as the sum of partial fractions.
    3. Determine the unknown coefficients.
    4. Integrate each term separately. Example:
  • Evaluate ∫ (5x + 3) / (x^2 + 4x + 3) dx. Solution:
  • Factoring the denominator, we have (x + 1)(x + 3).
  • Expressing the fraction as partial fractions, we have ∫ [A / (x + 1)] + [B / (x + 3)] dx.
  • Determining the coefficients, we find A = 2 and B = 1.
  • Integrating each term separately, we get 2 ln|x + 1| + ln|x + 3| + C.

Integration of Rational Functions with Repeated Linear Factors

  • Integrating rational functions with repeated linear factors involves partial fraction decomposition.
  • The steps include:
    1. Factorize the denominator, considering repeated factors.
    2. Express the fraction as the sum of partial fractions.
    3. Determine the unknown coefficients.
    4. Integrate each term separately. Example:
  • Evaluate ∫ (x^2 - 3x + 2) / (x^3 - 6x^2 + 11x - 6) dx. Solution:
  • Factoring the denominator, we have (x - 1)^2 (x - 2).
  • Expressing the fraction as partial fractions, we have `