In previous sections in the ‘Introduction to Integration Series’, we have covered the basics of finding the definite integral of a function f(x). In those sections, we were given the lower and upper bounds of integration, which helped us to find a number – the area under the curve.
In this section, we will take the integral of functions without the limits of integration. The result of the indefinite integral of the function is called the antiderivative, or indefinite integral.
Definition: The Antiderivative
The Antiderivative of a function, F(x), is a function which, if differentiated, becomes f(x): F’(x) = f(x). The process of finding the integral of a function is called Antidifferentiation, taking the indefinite integral, or integration. Differentiating and integrating are the opposites of each other. Conversely, if we take the integral of a function f(x), we obtain the antiderivative F(x): Where F(x) is the antiderivative and C is the constant of integration. |
You may be wondering, “Why do we add the C at the end of the antiderivative?” Let’s do some examples of differentiating and anti-differentiating to see why we must do so.
Consider the following functions:
- F(x) = x^{2}
- F(x) = x^{2}+3
- F(x) = x^{2}+ 23
- F(x) = x^{2}+ 151
The derivative of each of those functions are:
- f(x) = 2x
- f(x) = 2x
- f(x) = 2x
- f(x) = 2x
Notice that even though the original functions contained a constant in the end, the result of the differentiation is 2x (the derivative of a constant is zero).
If we antidifferentiate e, f, g, and h, we can get either x^{2}, x^{2}+3, x^{2}+ 23, x^{2}+ 151, or any other constant after the x^{2} term. However, how would we know the value of the constant? We don’t know the value of the constant. Hence, we will label the constant a variable C.
In conclusion, the anti-derivative represents a family of functions, written as:
Where the ∫ symbol is the integration symbol, F(x) is the integrand, ‘x’ is the variable of integration and C is the constant of integration.
Below is a summary of what we have learned so far:
- Indefinite integrals do not require limits of integration.
- The antiderivative F(x) also has a constant C, which can be positive or negative.
- An indefinite integral represents a family of functions; A definite integral gives you a numerical value (area under the curve).
Before we solve some examples for integration, we need to familiarize ourselves with some basic integration rules and properties.
In derivative calculus, we learned the basic properties and rules for differentiating certain functions. The properties and rules for integrating functions can be found directly from the derivative rules. Below is a table that lists the basic integration rules.
Rules | Function | Integral |
Constant Rule | ∫a dx | ax + C |
Multiplication by constant | ∫cf(x) dx | c∫f(x) dx = c* F(x) + C |
Power Rule (n≠-1) | ∫x^{n} dx | [x^{n+1} / (n+1)] + C |
Sum Rule | ∫(f + g) dx | ∫f dx + ∫g dx |
Difference Rule | ∫(f – g) dx | ∫f dx – ∫g dx |
Below is a list of integral of the most common functions. The integral of each of these functions will be considered more in detail in separate articles.
Common Functions | Function | Integral |
Constant | ∫a dx | ax + C |
Variable | ∫x dx | x^{2}/2 + C |
Square | ∫x^{2} dx | x^{3}/3 + C |
Reciprocal | ∫(1/u) dx
u = linear function inside denominator |
(1/u’)* ln|u| + C |
Exponential | ∫e^{u} dx
u = linear expression in exponent |
(1/u’)*e^{u} + C |
∫a^{u} dx
u = linear function inside exponent |
a^{u}/(u’*ln(a)) + C | |
Trigonometry (u in radians) | ∫cos(u) dx
u = linear expression inside parentheses |
(1/u’)*sin(u) + C |
∫sin(u) dx
u = linear expression inside parentheses |
-(1/u’)*cos(u) + C | |
∫sec^{2}(u) dx
u = linear expression inside parentheses |
(1/u’)*tan(u) + C |
Let’s go over the rules and the derivative of the most common functions in detail.
Constant Rule
Example 1: Find the integral of ∫3dx
Solution: The constant is 3. Therefore,
∫3dx= 3x + C
Example 2: Find the integral of ∫29dx
Solution: The constant is 29. Therefore,
∫29dx= 29x + C
For any function f(x) = a, the anti-derivative is F(x) = ax + C
Power Rule
Example 3: Find the integral of ∫x^{3}dx
Solution: The exponent is n = 3. Therefore,
Example 4: Find the integral of ∫x^{6}dx
Solution: The exponent is n = 6. Therefore,
Example 5: Find the integral of
Solution: The function can be rewritten as x^{1/2} .The exponent is n = ½. Therefore,
Example 6: Find the integral of
Solution: The function can be rewritten as x^{5/4} . The exponent is n = 5/4. Therefore,
For any function f(x) = x^{n}, the anti-derivative is
Multiplication by a Constant
Example 7: Find the integral of ∫3x^{6}dx
Solution: The constant is 3, so factor the 3:
∫3x^{6}dx
Now we can focus on the integrand x^{6} . We can use the power rule. The exponent is n = 6. Therefore,
Example 8: Find the integral of ∫3sin(x+1)dx
Solution: The constant is 3, so factor the 3 outside the integrand:
3∫sin(x+1)dx
Now we can focus on the integrand sin(x + 1) . Since the inside function is a linear function, we can use the integral table of common functions to see that
u = x +1
u’ = 1
Therefore,
Example 9: Find the integral of
Solution: The constant is 8, so factor the 8 from the integrand:
Now we can focus on the integrand . We can use the integral table of common functions to see that . However, keep in mind that this integral will work only if the inside function u is a polynomial of degree 1 (a linear function – mx+b):
u = 4x +2
Therefore,
Sum and Difference Rule
Example 10: Find the antiderivative of F (x) = 2x^{2}+ 3x
Solution:
Let’s break up the integral into 2 separate integrals. Factor each constant from each integral:
2 ∫ x^{2} dx + 3 ∫ x dx
Next, let’s integrate x^{2 }and x separately. Don’t forget to add the constant of integration C:
Afterwards, multiply their constants together:
Example 11: Find the antiderivative of F (x) = 6x^{4}+5x^{4}+3x^{2}
Solution:
Let’s break up the integral into 3 separate integrals. Factor each constant from each integral:
6 ∫ x^{4} dx + 5∫x^{3} dx + 3∫x^{2} dx
Next, let’s integrate x^{4},x^{3}and x^{2} separately. Don’t forget to add the constant of integration C:
Afterwards, multiply their constants together:
Example 12: Find the antiderivative of f (x) =
Solution:
Let’s break the rational function into three separate terms:
Now we can take the integral of each term and add them:
Let’s factor the numerator and denominator from each integral:
Simplifying each integrand and rational constant, using the quotient rule for exponents:
Now we can integrate by using the power rule for integration:
Example 13: Find the antiderivative of f(x) =
Solution:
Let’s take the integral of each term:
Evaluate each definite integral:
Example 14: Find the antiderivative of f (x) = 3^{3x}-sec^{2}(3x+4)-9x^{3/5}
Solution:
We see that the first term is an exponential function, the second term is a variable term with an exponent. Let’s take the integral of each term using the table of integrals:
Summary of Section
The antiderivative, also known as the indefinite integral, of a function f(x) is a differentiable function F(x) whose derivative is equal to the original function f(x): F’(x) = f(x) The process of solving for antiderivatives is called antidifferentiation or taking the definite integral. Differentiation and integration are the opposite of each other. The result of the indefinite integral of f(x) is a family of antiderivatives: ∫f(x)dx = F(x) + C Where F(x) is the anti-derivative and C is the constant of integration. A table of antiderivatives can be used to compute the indefinite integral of simple functions, such as constants, powers, trigonometric functions, exponentials, and reciprocal functions. |
References:
https://wwwi.scottsdalecc.edu/softchalk/math/MAT212/Module_5/def_int/def_int_print.html
https://slideplayer.com/slide/12935447/
https://www.mathsisfun.com/calculus/integration-rules.html#examples
http://scidiv.bellevuecollege.edu/dh/Calculus_all/CC_4_6_FindingAntider.pdf
http://www.opentextbookstore.com/appcalc/Chapter3-3.pdf
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