Antiderivatives

Derivatives are the way to measure the rate of change, given a function f(x), the derivatives for that function calculate the rate of change of the value of that function at a particular point. Sometimes there are situations where the derivative is available, but no information is given about the function. In that case, anti-derivatives come to the rescue. It becomes essential to understand the concept and the methods for using anti-derivatives to generate the functions from their derivatives. Let’s look at this concept in detail. 

Anti-Derivative

Let’s consider a function f(x) = x², on differentiating this function, the output is another function g(x) = 2x. Anti-Derivatives are supposed to generate the function f(x) given another function g(x). Notice that while calculating the derivative, the exponent on the variable “x” decreased by 1, so in case of the reverse process the exponent will be increased. Also, in the case of another function h(x) = x² + 2, the derivative of this function is still g(x). Since constant cancels out while differentiating, we can conclude that for any function f(x), there can infinitely many anti-derivative functions with different values of the constants. This means, 

For, g(x) = 2x the anti-derivative will be, 

f(x) = x² + C, where C is a constant. 

A function F is called anti-derivative of the function, if 

F'(x) = f(x) 

For all x in the domain of f. 

If F(x), is the anti-derivative of f(x) then the most general anti-derivative of f(x) is called indefinite integral. 

∫f(x)dx = F(x) + C, C is any constant. 

Here the symbol ∫ denotes the anti-derivative operator, it is called indefinite integrals.

Properties of Indefinite integrals

There are some important properties that come in handy while calculating the anti-derivatives for the functions. 

1. ∫kf(x)dx = k ∫f(x)dx, here “k” is any constant.
2. ∫-f(x)dx = -∫f(x)dx, This property can be thought as a special case of the previous property with k = -1.
3. ∫(f(x) + g(x))dx = ∫f(x)dx + ∫g(x)dx. This property shows that the integral of sum of two functions is equal to sum of integrals of those two functions.

Note: Keep in my mind that property 3 is only applicable for subtraction and summation. It cannot and should not be extended up to multiplication and division operations. 

Computing Indefinite Integrals

It is not always possible to just guess the integral of any function by thinking of the reverse differentiation process. A formal approach or a formula is necessary for calculating the indefinite integrals. 

Consider a function of the form xⁿ, the anti-derivative for this function is given by:

 except when n = -1. 

So, the general rule for integrating such functions is to increase the exponent value by 1 and then divide the function by that value. The table below represents some standard functions and their integrals. 

Function	Integral
sin(x)	-cos(x) + C
cos(x)	sin(x) + C
sec2(x)	tan(x) + C
ex	ex
	ln(x) + C

Let us see some sample problems with these concepts. 

Sample Problems

Question 1: Find the integral for the given function, 

f(x) = 2x + 3

Solution: 

Using the integral formula for the functions of the type of xⁿ. 

Given, f(x) = 2x + 3

Splitting the function using the property 3. 

⇒

⇒

⇒

⇒ x² + 3x + C

Question 2: Find the integral for the given function, 

f(x) = x² – 3x

Solution: 

Using the integral formula for the functions of the type of xⁿ. 

Given, f(x) = x² – 3x

Splitting the function using the property 3. 

⇒

⇒

Question 3: Find the integral for the given function, 

f(x) = x³ + 5x² + 6x + 1

Solution: 

Using the integral formula for the functions of the type of xⁿ. 

Given, f(x) = x³ + 5x² + 6x + 1

Splitting the function using the property 3. 

⇒

⇒

Question 4: Find the integral for the given function, 

f(x) = sin(x) – cos(x)

Solution: 

Using the integral formula for the trigonometric function given below.

Given, f(x) = sin(x) – cos(x)

Splitting the function using the property 3. 

⇒

⇒

Question 5: Find the integral for the given function, 

f(x) = 2sin(x) + sec²(x) + 7e^x

Solution: 

Using the integral formula for the trigonometric function given below.

Given, f(x) = 2sin(x) + sec²(x) + 7e^x

Splitting the function using the property 3. 

⇒

⇒

⇒

Question 6: Find the integral for the given function, 

f(x) = 

Solution: 

Using the integral formula for the functions of the type of xⁿ. 

Given, f(x) = 

Splitting the function using the property 3. 

⇒

⇒ x – 3ln(x) + C

Question 7: Find the integral for the given function, 

f(x) = x² – 4x + 4

Solution: 

Using the integral formula for the functions of the type of xⁿ. 

Given, f(x) = x² – 4x + 4

Splitting the function using the property 3. 

⇒