Indefinite Integrals

Integration is the inverse process of differentiation. Instead of finding the derivative of a function, we start with a derivative and work backwards to find the original function. This is also known as anti-differentiation.

Note that a constant added to any function doesn't affect its derivative, which is why the result of integration always includes a constant C, known as the constant of integration.

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

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 = xⁿ⁺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
• ∫ sec x dx = tan x + C

Finding Indefinite Integral

Various different methods are used to calculate the indefinite integrals,

• 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.

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

Properties of Indefinite Integrals

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

Property of Sum

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

Property of Difference

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

Property of Constant Multiple

∫ 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 = k_{₁∫ f₁(x)dx} + k_{₂∫ f₂(x)dx} + ... + kₙ∫ fₙ(x)dx

Indefinite Integral vs Definite Integral

Solved 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) = 2eˣ. .

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}

Related Articles:

• Calculus
• Integral Calculus
• Application of Integrals
• Integration Methods