Integration Rules
Integration rules: Integration is used to find many useful parameters or quantities like area, volumes, central points, etc., on a large scale. The most common application of integration is to find the area under the curve on a graph of a function.
To work out the integral of more complicated functions than just the known ones, we have some integration rules. These rules can be studied below. Apart from these rules, there are many integral formulas that substitute the integral form.
Integration Rules of Basic Functions
The integration rules are defined for different types of functions. Let us learn here the basic rules for integration of the som common functions, such as:
- Constant
- Variable
- Square
- Reciprocal
- Exponential
- Trigonometry
Integration of Constant
Integration of constant function say ‘a’ will result in:
∫a dx = ax + C
Example:
∫4 dx = 4x + C
Integration of Variable
If x is any variable then;
∫x dx = x^{2}/2 + C
Integration of Square
If the given function is a square term, then;
∫x^{2} dx = x^{3}/3
Integration of Reciprocal
If 1/x is a reciprocal function of x, then the integration of this function is:
∫(1/x) dx = ln|x| + C (Natural log of x)
Integration of Exponential Function
The different rules for integration of exponential functions are:
- ∫e^{x} dx = e^{x} + C
- ∫a^{x} dx = a^{x}/ln(a) + C
- ∫ln(x) dx = x ln(x) − x + C
Integration of Trigonometric Function
- ∫cos(x) dx = sin(x) + C
- ∫sin(x) dx = -cos(x) + C
- ∫sec^{2}(x) dx = tan(x) + C
Important Integration Rules
The important rules for integration are:
- Power Rule
- Sum Rule
- Different Rule
- Multiplication by Constant
- Product Rule
Power Rule of Integration
As per the power rule of integration, if we integrate x raised to the power n, then;
∫x^{n} dx = (x^{n+1}/n+1) + C
By this rule the above integration of squared term is justified, i.e.∫x^{2} dx. We can use this rule, for other exponents also.
Example: Integrate ∫x^{3}dx.
∫x^{3} dx = x^{(3+1)}/(3+1) = x^{4}/4
Sum Rule of Integration
The sum rule explains the integration of sum of two functions is equal to the sum of integral of each function.
∫(f + g) dx = ∫f dx + ∫g dx
Example: ∫(x + x^{2} )dx
= ∫x dx + ∫x^{2 }dx
= x^{2}/2 + x^{3}/3 + C
Difference Rule of Integration
The difference rule of integration is similar to the sum rule.
∫(f – g) dx = ∫f dx – ∫g dx
Example: ∫(x – x^{2} )dx
= ∫x dx – ∫x^{2 }dx
= x^{2}/2 – x^{3}/3 + C
Multiplication by Constant
If a function is multiplied by a constant then the integration of such function is given by:
∫cf(x) dx = c∫f(x) dx
Example: ∫2x.dx
= 2∫x.dx
=2 x^{2}/2 + C
= x^{2} + C
Apart from the above-given rules, there are two more integration rules:
Integration by parts
This rule is also called the product rule of integration. It is a special kind of integration method when two functions are multiplied together. The rule for integration by parts is:
∫ u v da = u∫ v da – ∫ u'(∫ v da)da
Where
- u is the function of u(a)
- v is the function of v(a)
- u’ is the derivative of the function u(a)
Integration by Substitution
Integration by substitution is also known as “Reverse Chain Rule” or “u-substitution Method” to find an integral.
The first step in this method is to write the integral in the form:
∫ f(g(x))g'(x)dx
Now, we can do a substitution as follows:
g(x) = a and g'(a) = da
Now substitute the equivalent values in the above form:
∫ f(a) da
Once you integrate the above form, finally substitute the original values.
Learn more about: Integration by substitution
Solved Examples
Question 1: What is ∫ 8 a^{3} da?
Solution: We can take 8 out of integral,
∫ 8 a^{3} da = 8 ∫ a^{3} da
= 8 a^{4} / 4 + C
= 2 a^{4 } + C
Question 2: What is ∫ 4 a^{3} da?
Solution: We can take 8 out of integral,
∫ 4 a^{3} da = 4 ∫ a^{3} da
= 4 a^{4} / 4 + C
= a^{4 } + C
Question 3: What is ∫ Cos a + a da ?
Solution: ∫ Cos a + a da = ∫ Cos a da + ∫ a da
= sin a + a^{2} /2 + C
Question 4: What is ∫ Sin a + a da ?
Solution: ∫ Sin a + a da = ∫ Sin a da + ∫ a da
= – Cos a + a^{2} /2 + C
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Learn more on Integration | |
Integration by Substitution | Methods of Integration |
Proofs of Integration Formula | Integration by Parts |
Frequently Asked Questions – FAQs
What is the power rule of integration?
∫x^{n} dx = x^{n+1}/n+1 + C
What is the Sum rule of integration?
∫(f + g) dx = ∫f dx + ∫g dx
What is the product rule of integration?
∫u v dx = u∫v dx −∫u’ (∫v dx) dx
What is the value of integral of function when multiplied by a constant?
∫cf(x) dx = c∫f(x) dx
Thus, the constant will be removed out of an integral part.
What is the rule of integration for exponential function?
∫e^{x} dx = e^{x} + C