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 some 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 = x²/2 + C

Integration of Square

If the given function is a square term, then;

∫x² dx = x³/3

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
• ∫sec²(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;

∫xⁿ dx = (xⁿ⁺¹/n+1) + C

By this rule the above integration of squared term is justified, i.e.∫x² dx. We can use this rule, for other exponents also.

Example: Integrate ∫x³dx.

∫x³ dx = x⁽³⁺¹⁾/(3+1) = x⁴/4

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 + x² )dx 

= ∫x dx + ∫x² dx

= x²/2 + x³/3 + 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 – x² )dx 

= ∫x dx – ∫x² dx

= x²/2 – x³/3 + 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 x²/2 + C

= x² + 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 a³ da?

Solution: We can take 8 out of integral,

∫ 8 a³ da = 8 ∫ a³ da

= 8 a⁴ / 4 + C

= 2 a⁴ + C

Question 2: What is ∫ 4 a³ da?

Solution: We can take 4 out of integral,

∫ 4 a³ da = 4 ∫ a³ da

= 4 a⁴ / 4 + C

=  a⁴ + C

Question 3: What is ∫ (Cos a + a) da ?

Solution: ∫ (Cos a + a) da = ∫ Cos a da + ∫ a da

= sin a + a² /2 + C

Question 4: What is ∫ (Sin a + a) da ?

Solution: ∫ (Sin a + a) da = ∫ Sin a da + ∫ a da

= – Cos a + a² /2 + C

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