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.

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:

Integration of constant function say ‘a’ will result in:

∫a dx = ax + C

Example:

∫4 dx = 4x + C

If x is any variable then;

∫x dx = x2/2 + C

If the given function is a square term, then;

∫x2 dx = x3/3

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)

The different rules for integration of exponential functions are:

The important rules for integration are:

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

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

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

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:

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

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

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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