Integration Formulas

Integration formulas are the basic rules used to find the integral (antiderivative) of a function. They help solve different types of integration problems involving algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions. By using these formulas, complex integrals can be evaluated more easily and efficiently.

Note:

• d/dx [∫f(x) dx] = f(x)
• ∫k. f(x) dx = k ∫f(x) dx , where k is constant
• ∫{f(x) ± g(x)} dx = ∫f(x) dx  ± ∫g(x) dx

Basic Integration Formulas

Integration Formulas of Trigonometric Functions

Integration Formulas of Trigonometric functions are used to solve the integral equations involving Trigonometric functions.

• ∫ cos x dx = sin x + C
• ∫ sin x dx = -cos x + C
• ∫ sec²x dx = tan x + C
• ∫ cosec²x dx = -cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec x cot x dx = -cosec x + C
• ∫ tan x dx = log |sec x| + C
• ∫ cot x dx = log |sin x| + C
• ∫ sec x dx = log |sec x + tan x| + C
• ∫ cosec x dx = log |cosec x - cot x| + C

Integration Formulas of Inverse Trigonometric Functions

Various Integration Formulas of Inverse Trigonometric Functions, which are used to solve integral questions, are given below:

• ∫1/√(1 - x²) dx = sin^-1x + C
• ∫ -1/√(1 - x²) dx = cos^-1x + C
• ∫1/(1 + x²) dx = tan^-1x + C
• ∫ -1/(1 + x²) dx = cot^-1x + C
• ∫ 1/x√(x² - 1) dx = sec^-1x + C
• ∫ -1/x√(x² - 1) dx = cosec^-1 x + C

Advanced Integration Formulas

Some other advanced integration formulas that are of high importance for solving integrals are discussed below:

• ∫1/(x² - a²) dx = 1/2a ln|(x - a)(x + a| + C
• ∫ 1/(a² - x²) dx =1/2a ln|(a + x)/(a - x)| + C
• ∫1/(x² + a²) dx = 1/a tan^-1x/a + C
• ∫1/√(x² - a²)dx = log |x +√(x² - a²)| + C
• ∫ √(x² - a²) dx = x/2 √(x² - a²) -a²/2 log |x + √(x² - a²)| + C
• ∫1/√(a² - x²) dx = sin^-1 x/a + C
• ∫√(a² - x²) dx = x/2 √(a² - x²) dx + a²/2 sin^-1 x/a + C
• ∫1/√(x² + a² ) dx = log |x + √(x² + a²)| + C
• ∫ √(x² + a² ) dx = x/2 √(x² + a² )+ a²/2 log |x + √(x² + a²)| + C

Integration by Parts Formula

Integration by Parts is applied when the given function is easily described as the product of two functions. The integration by Parts formula used in mathematics is given below,

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

Example: Calculate ∫ xe^x dx

Solution:

∫ xe^x dx is of the form ∫ f(x) g(x) dx

let f(x) = x and g(x) = e^x

we know that, ∫ f(x) g(x) dx = f(x) ∫g(x) dx - ∫ (f'(x) ∫g(x) dx) dx + C

∫ xe^x dx = x ∫e^x dx - ∫( 1 ∫e^x dx) dx+ c

= xe^x - e^x + c

Integration by Substitution Formula

Integration by Substitution is applied when a function is a function of another function. i.e. let I = ∫ f(x) dx, where x = g(t) such that dx/dt = g'(t), then dx = g'(t)dt

Now, I = ∫ f(x) dx = ∫ f(g(t)) g'(t) dt

Example: Evaluate ∫ (4x +3)³ dx

Solution:

Let u = (4x+3) ⇒ du = 4 dx

∫ (4x +3)³ dx 
= 1/4 ∫(u)³ du
= 1/4. u⁴ /4 
= u⁴ /16
= (4x +3)⁴/16

Integration by Partial Fractions Formula

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, such that the degree of P(x) is less than the (<) the  degree of Q(x), then the fraction P(x)/Q(x) is written as

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

Now, the integration of R(x) + P₁(x)/ Q(x) is easily calculated using the formulas discussed above.

➢Practice: Solved Examples