Introduction to Integration

• Integration is the inverse process of differentiation.

Basic Concepts

• Integration finds the function given its derivative.
• If the derivative of F(x) is f(x), then the integral of f(x) is F(x) + C, where C is the constant of integration.
• Represented as ∫f(x) dx = F(x) + C, where ∫ is the integral sign, f(x) is the integrand, dx indicates integration with respect to x, F(x) is the antiderivative, and C is the constant of integration.

Types of Integrals

• Indefinite Integrals: These integrals do not have specific limits of integration, resulting in a general function plus a constant (C).
• Definite Integrals: These integrals have upper and lower limits, resulting in a numerical value.

Basic Integration Formulas

• ∫x^n dx = (x^(n+1))/(n+1) + C, for n ≠ -1
• ∫(1/x) dx = log|x| + C
• ∫e^x dx = e^x + C
• ∫a^x dx = (a^x)/log(a) + C
• ∫sin(x) dx = -cos(x) + C
• ∫cos(x) dx = sin(x) + C
• ∫sec^2(x) dx = tan(x) + C
• ∫cosec^2(x) dx = -cot(x) + C
• ∫sec(x)tan(x) dx = sec(x) + C
• ∫cosec(x)cot(x) dx = -cosec(x) + C
• ∫(1/√(1-x^2)) dx = sin⁻¹(x) + C
• ∫(-1/√(1-x^2)) dx = cos⁻¹(x) + C
• ∫(1/(1+x^2)) dx = tan⁻¹(x) + C
• ∫(-1/(1+x^2)) dx = cot⁻¹(x) + C
• ∫(1/(x√(x^2-1))) dx = sec⁻¹(x) + C
• ∫(-1/(x√(x^2-1))) dx = cosec⁻¹(x) + C

Properties of Indefinite Integrals

• ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
• ∫k*f(x) dx = k∫f(x) dx, where k is a constant.

Methods of Integration

• Integration by Substitution: Used to simplify integrals by substituting a function with a new variable, simplifying the integrand.
  • Choose a suitable substitution u = g(x).
  • Find du/dx and express dx in terms of du.
  • Substitute u and du into the integral.
  • Evaluate the new integral with respect to u.
  • Replace u with g(x) to get the final answer in terms of x.
• Integration by Parts: Used to integrate products of functions, based on the formula ∫u dv = uv - ∫v du.
  • Choose u and dv from the integrand.
  • Find du and v.
  • Apply the formula ∫u dv = uv - ∫v du.
  • Evaluate the new integral ∫v du.
  • The choice of u and dv is guided by the ILATE rule (Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential), where the function that comes first is chosen as u.
• Integration by Partial Fractions: Used to integrate rational functions by breaking them into simpler fractions.
  • Factorize the denominator of the rational function.
  • Express the rational function as a sum of partial fractions based on the factors of the denominator.
  • Determine the constants in the numerators of the partial fractions by equating coefficients or using specific values of x.
  • Integrate each partial fraction separately.
• Integration using Trigonometric Identities: Simplifies integrals using trigonometric identities.
  • Rewrite the integrand using appropriate trigonometric identities.
  • Integrate the simplified expression.

Definite Integrals

• A definite integral has the form ∫[a to b] f(x) dx, where a is the lower limit and b is the upper limit.
• The definite integral represents the area under the curve of f(x) from x = a to x = b.
• ∫[a to b] f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x).

Properties of Definite Integrals

• ∫[a to b] f(x) dx = -∫[b to a] f(x) dx
• ∫[a to a] f(x) dx = 0
• ∫[a to b] f(x) dx = ∫[a to c] f(x) dx + ∫[c to b] f(x) dx
• ∫[a to b] f(x) dx = ∫[a to b] f(t) dt
• ∫[0 to a] f(x) dx = ∫[0 to a] f(a - x) dx
• ∫[-a to a] f(x) dx = 2∫[0 to a] f(x) dx, if f(x) is an even function.
• ∫[-a to a] f(x) dx = 0, if f(x) is an odd function.
• ∫[0 to 2a] f(x) dx = 2∫[0 to a] f(x) dx, if f(2a - x) = f(x)
• ∫[0 to 2a] f(x) dx = 0, if f(2a - x) = -f(x)

Applications of Integrals

• Finding the area under a curve: The area under the curve y = f(x) from x = a to x = b is given by ∫[a to b] f(x) dx.
• Finding the area between two curves: The area between two curves y = f(x) and y = g(x) from x = a to x = b is given by ∫[a to b] |f(x) - g(x)| dx.
• Finding volumes of solids of revolution:
  • Disk Method: Volume = π∫[a to b] [f(x)]^2 dx
  • Washer Method: Volume = π∫[a to b] ([f(x)]^2 - [g(x)]^2) dx
• Determining displacement and distance given velocity functions.

Special Integrals

• ∫(1/√(a^2 - x^2)) dx = sin⁻¹(x/a) + C
• ∫(1/(a^2 + x^2)) dx = (1/a)tan⁻¹(x/a) + C
• ∫(1/(x√(x^2 - a^2))) dx = (1/a)sec⁻¹(x/a) + C
• ∫(1/(a^2 - x^2)) dx = (1/2a)log|(a+x)/(a-x)| + C
• ∫(1/(x^2 - a^2)) dx = (1/2a)log|(x-a)/(x+a)| + C
• ∫√(a^2 - x^2) dx = (x/2)√(a^2 - x^2) + (a^2/2)sin⁻¹(x/a) + C
• ∫√(x^2 + a^2) dx = (x/2)√(x^2 + a^2) + (a^2/2)log|x + √(x^2 + a^2)| + C
• ∫√(x^2 - a^2) dx = (x/2)√(x^2 - a^2) - (a^2/2)log|x + √(x^2 - a^2)| + C
• ∫e^(ax)cos(bx) dx = (e^(ax)/(a^2 + b^2))(a cos(bx) + b sin(bx)) + C
• ∫e^(ax)sin(bx) dx = (e^(ax)/(a^2 + b^2))(a sin(bx) - b cos(bx)) + C

Trigonometric Integrals

• Integrals involving powers of sine and cosine: Use trigonometric identities to simplify the integral.
• Integrals involving powers of tangent and secant: Use trigonometric identities to simplify the integral.
• Integrals involving products of sines and cosines with different arguments: Convert products to sums or differences using trigonometric identities.

Common Mistakes

• Forgetting the constant of integration (C) in indefinite integrals.
• Incorrectly applying the integration by parts formula.
• Making errors in algebraic manipulation during substitution.
• Not simplifying the integrand before attempting integration.
• Incorrectly applying limits of integration in definite integrals.