Examples on Antiderivatives

Example 1: Find the integral for the given function, 

f(x) = 2x + 3

Solution: 

Using Integral Formula,

Given,

f(x) = 2x + 3

Splitting the function

⇒

⇒

⇒

⇒ x² + 3x + C

Example 2: Find the integral for the given function, 

f(x) = x² – 3x

Solution:

Using Integral Formula,

Given,

f(x) = x² – 3x

Splitting the function

⇒

⇒

Example 3: Find the integral for the given function, 

f(x) = x³ + 5x² + 6x + 1

Solution: 

Using Integral Formula,

Given,

f(x) = x³ + 5x² + 6x + 1

Splitting the function

⇒

⇒

Example 4: Find the integral for the given function, 

f(x) = sin(x) – cos(x)

Solution: 

Using Integral Formula,

Given,

f(x) = sin(x) – cos(x)

Splitting the function

⇒

⇒

Example 5: Find the integral for the given function, 

f(x) = 2sin(x) + sec²(x) + 7e^x

Solution: 

Given,

f(x) = 2sin(x) + sec²(x) + 7e^x

Splitting the function

⇒

⇒

⇒

Example 6: Find the integral for the given function, 

f(x) = 

Solution: 

Using Integral Formula,

Given,

f(x) = 

Splitting the function

⇒

⇒ x – 3ln(x) + C

Example 7: Find the integral for the given function, 

f(x) = x² – 4x + 4

Solution: 

Using Integral Formula,

Given,

f(x) = x² – 4x + 4

Splitting the function

⇒

Antiderivatives: The Antiderivative of a function is the inverse of the derivative of the function. Antiderivative is also called the Integral of a function. Suppose the derivative of a function d/dx[f(x)] is F(x) + C then the antiderivative of [F(x) + C] dx of the F(x) + C is f(x). An example explains this if d/dx(sin x) is cos x then, the antiderivative of cos x, given as ∫(cos x) dx is sin x.

Antiderivative of any function is used for various purposes, to give the area of the curve, to find the volume of any 3-D curve, and others. In this article, we will learn about, Antiderivatives, Antiderivatives Formulas, Antiderivatives rules, and others in detail.