Indefinite Integrals

Integrals are also known as anti-derivatives as integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and are required to calculate the function from the derivative. This process is called integration or anti-differentiation.

If f(x) is a continuous function on an interval I, an indefinite integral of f is a function F(x) such that:

F′(x) = f(x) for all x ∈ I

Consider a function f(x) = sin(x), the derivative of this function if f'(x) = cos(x). So, the integration of f'(x) should give back the function f(x). Notice that for every function f(x) = sin (x) + C, the derivative is the same because the constant becomes zero after differentiation.

This relationship is expressed using the integral symbol without upper and lower limits:

∫f(x) dx = F(x) + C

Where ∫ is the symbol for integral.

The table below represents the symbols and meanings related to integrals.

Formulas for Indefinite Integrals

There are certain formulas and rules which when kept in mind, help us simplify the calculating and do it fast. Some of these formulas are:

• ∫ 1 dx = x + C
• ∫ P dx = Px + C
• ∫ xⁿ dx = x^{n + 1}/ (n + 1) + C
• ∫ e^x dx = e^x + C
• ∫ a^x dx = a^x / ln a + C
• ∫1/x dx = ln |x| + C
• ∫ cos x dx = sin x + C
• ∫ sin x dx = -cos x + C
• ∫ sec²x dx = tan x + C

How to Find Indefinite Integral

Various different methods are used to calculate the indefinite integrals are,

• Normal indefinite integrals are solved using direct integration formulas.
• Integrals with rational functions are solved using the partial fractions method.
• Indefinite integrals can be solved using the substitution method.
• Integration by parts is used to solve the integral of the function where two functions are given as a product.

Let's consider an example for better understanding.

Example: Find the indefinite integral ∫ x³ cos x⁴ dx

Solution:

Using the substitution method. 

Let x⁴ = t
⇒ 4x³ dx = dt

Now, ∫ x³ cos x⁴ dx
= 1/4∫cos t dt
= 1/4 (sin t) + C
= 1/4 sin (x⁴ ) + C

Read More about Integration Methods.

Properties of Indefinite Integrals

Indefinite integrals have various properties some of the various properties of Indefinite Integral are,

Property of Sum

The property of the Sum of Indefinite Integral is,

∫ [f(x) + g(x)]dx = ∫ f(x)dx + ∫ g(x)dx

Property of Difference

The property of the Difference of Indefinite Integral is,

∫ [f(x) × g(x)]dx = ∫ f(x)dx × ∫ g(x)dx

Property of Constant Multiple

The property of the Constant Multiple of Indefinite Integral is,

∫ k f(x)dx = k∫ f(x)dx

Some of the other properties of the indefinite integral are,

• ∫ f(x) dx = ∫ g(x) dx if ∫ [f(x) - g(x)]dx = 0
• ∫ [k₁f₁(x) + k₂f₂(x) + ...+k_nf_n(x)]dx = k₁∫ f₁(x)dx + k₂∫ f₂(x)dx + ... + k_n∫ f_n(x)dx

Difference Between Indefinite Integral and Definite Integral

Some of the key differences between indefinite and definite integrals are:

Article Related:

• Calculus
• Integral Calculus
• Application of Integrals

Indefinite Integrals Examples

Example 1: Find the integral for the given function f(x), f(x) = sin(x) + 1

Solution: 

Given f(x) = sin(x) + 1

sin(x) is a standard function, and it's anti-derivative is,

∫ f(x)dx
= ∫ (sin(x) + 1)dx
= \int sin(x)dx  + \int 1dx
= -cos(x) + x + C

Example 2: Find the integral for the given function f(x), f(x) = 2e^x 

Solution: 

Given f(x) = 2e^x 

e^x is a standard function, and it's anti-derivative is,

= \int f(x)dx
= \int 2e^xdx

Using the property 1 mentioned above, 

= 2\int e^xdx
= 2e^x + C

Example 3: Find the integral for the given function f(x), f(x) = 5x^-2

Solution: 

Given f(x) = 5x^-2

Using reverse power rule

= \int f(x)dx
= \int 5x^{-2}dx

Using property 1 mentioned above, 

= 5\int x^{-2}dx
= \frac{-5}{x} + C

Example 4: Find the integral for the given function f(x), f(x) = sin(x) + 5cos(x)

Solution: 

Given f(x) = sin(x) + 5cos(x)

sin(x) and cos(x) are standard functions, and its integral is,

= \int f(x)dx
= ∫ (sin(x) + 5cos(x))dx
= \int sin(x)dx  + 5\int cos(x)dx
= -cos(x) + 5sin(x) + C

Example 5: Find the integral for the given function f(x), f(x) = 5x^-2 + x⁴ + x

Solution: 

Given f(x) = 5x^-2 + x⁴ + x

Using reverse power rule

= \int f(x)dx
= \int (5x{-2} + x^4  + x)dx
= \int (5x{-2} + x^4  + x)dx
= 5\int x^{-2}dx + \int x^4dx   + \int xdx
= \frac{-5}{x} + \frac{x^5}{5}   + \frac{x^2}{2}