Formulas of Integration

Integration is a fundamental calculus concept where one has to find a function corresponding to its derivative. It is a method of combining parts to find the whole. Integration formulas can be used to solve different types of integration problems in algebra, trigonometry, exponential functions, logarithms, and inverse trigonometry. They consist of trigonometric ratios, product of functions, inverse trigonometric functions and advanced integration formulas. It is a reverse process of differentiation. So, if d/dx (y) = z, then ∫zdx = y. The fundamental integration formula, which serves as the basis for serving all integration formulas is

∫ f'(x)dx = f(x) + C

Classification of Integration Formulas

Integral Formulas are classified based on the following functions:

• Rational functions
• Irrational functions
• Trigonometric functions
• Inverse trigonometric functions
• Hyperbolic functions
• Exponential functions
• Logarithmic functions
• Gaussian functions
• Inverse hyperbolic functions

Integrations by Parts and Substitution Formulas

Integration by Parts Formula

The integration by parts formula is applied when the given function can be described as the product of two functions:

∫ f(x)g(x) dx = f(x)∫ g(x) dx - ∫ (∫f'(x)g(x) dx) dx + C

Integration by Substitution Formula

Integration by substitution is applied when a function is a function of another function. That is, if I = ∫ f(x) dx, where x = g(t) such that dx/dt = g'(t), then dx = g'(t)dt:

I = ∫ f(x) dx = ∫ f(g(t)) g'(t) dt

Integral by Partial Fractions

The formula for Integration by Partial Fractions is used when the integral of P(x)/Q(x) is required and P(x)/Q(x) is an improper fraction:

P(x)/Q(x) = R(x) + P1(x)/ Q(x)

where:

• R(x) is a polynomial in x
• P1(x)/ Q(x) is a proper rational function

Definite and Indefinite Integration Formulas

The formula for Definite integral is used when the integration limit is given. In definite integration, the solution to the question is a constant value. Definite integration is solved as:

∫_a^b f(x) dx = F(b) - F(a)

The formula for Indefinite Integration Formula is used to solve the indefinite integration when the limit of integration is not given. In indefinite integration, we use the constant of the integration which is denoted by C:

∫ f(x) dx = F(x) + C

Basic Trigonometric Integral Formulas

∫ cosx dx = sinx + C

∫ sinx dx = -cosx + C

Powers of Trigonometric Functions

∫ sec2x dx = tanx + C

∫ csc2x dx = -cotx + C

Products of Trigonometric Functions

∫ secxtanx dx = secx + C

∫ cscxcotx dx = -cscx + C

Inverse Trigonometric Integrals

∫ tanx dx = ln|secx| + C

∫ cotx dx = ln|sinx| + C

∫ secx dx = ln|secx + tanx| + C

∫ cscx dx = ln|cscx - cotx| + C

Integration Formulas Involving Inverse Trigonometric Functions

Arcsine Function

∫ 1/(1 - x^2) dx = sin-1x + C

∫ -1/(1 - x^2) dx = cos-1x + C

Arctangent Function

∫ 1/(1 + x^2) dx = tan-1x + C

∫ -1/(1 + x^2) dx = cot-1x + C

Arcsecant Function

∫ 1/(x√(x^2 - 1)) dx = sec-1x + C

∫ -1/(x√(x^2 - 1)) dx = csc-1x + C

Examples of Integration Formulas

Example 1: Integration of ∫ 1/(x^2 + a^2) dx

Given: ∫ 1/(x^2 + a^2) dx

Formula used: ∫ 1/(x^2 + a^2) dx = (1/a) tan-1(x/a) + C

Solution: Applying the formula: ∫ 1/(x^2 + a^2) dx = (1/a) tan-1(x/a) + C

Example 2: Integration of ∫ √(x^2 - a^2) dx

Given: ∫ √(x^2 - a^2) dx

Formula used: ∫ √(x^2 - a^2) dx = (x/2) √(x^2 - a^2) - (a^2/2) log|x + √(x^2 - a^2)| + C

Solution: Applying the formula: ∫ √(x^2 - a^2) dx = (x/2) √(x^2 - a^2) - (a^2/2) log|x + √(x^2 - a^2)| + C

Example 3: Integration of ∫ 1/√(1 - x^2) dx

Given: ∫ 1/√(1 - x^2) dx

Formula used: ∫ 1/√(1 - x^2) dx = sin-1x + C

Solution: Applying the formula: ∫ 1/√(1 - x^2) dx = sin-1x + C

These examples illustrate the application of integration formulas to find antiderivatives of various functions. Each formula is specialized to handle different types of integrals, facilitating efficient problem-solving in calculus and beyond.