Do You Know The Important Integration Rules?

Integration is a method of bringing different parts to form a whole. Integration is simply the inverse process of differentiation. We use integration in mathematics to find areas, volumes, etc. The area of the region enclosed by the graph of functions is defined and calculated using integration.

Integration Rules of Basic Functions

We have some integration rules to find out the integral of more complicated functions than just the known ones. These rules can be read in the sections below. Besides these rules, there are a number of integral formulas that can be used to replace the integral form.

Different types of functions have different integration rules. Let's look at some basic integration rules for some basic functions, such as:

1. Constant Function

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

$\Rightarrow \int a~dx=ax+C$

For example:$\Rightarrow \int 7~dx=7x+C$

Where C is the integral constant.

2. Linear Variable Function

If x is any given variable then, we can write this as,

$\Rightarrow \int x~dx=\frac{x^2}{2}+C$

Where C is the integral constant.

3. Square Function

If the square function is given, then,

$\Rightarrow \int x^2~dx=\frac{x^3}{3}+C$

Where C is the integral constant.

4. Reciprocal Function

We know 1/x is a reciprocal function of x. The integration for this function can be written as:

$\Rightarrow \int \frac{1}{x}dx=\ln|x|+C$

Where C is the integral constant.

5. Exponential Function

The rules of integration of different exponential functions are given below:

$\Rightarrow \int e^x~dx=e^x+C$

$\Rightarrow \int a^x~dx=\frac{a^x}{\ln(a)}+C$

$\Rightarrow \int \ln(x)~dx=x\ln(x)-x+C$

Where C is the integral constant.

6. Trigonometric Function

$\Rightarrow \int \cos x~dx=\sin x+C$

$\Rightarrow \int \sin x~dx=-\cos x+C$

$\Rightarrow \int \sec^2 x~dx=\tan x+C$

Where C is the integral constant.

Important Integration Rules

As we discussed basic integration rules above, let’s also look at some of the important rules for integration below:

• Power Rule

• Sum Rule

• Different Rule

• Multiplication by Constant

• Product Rule

Power Rule

If we integrate x raised to the power n according to the power rule of integration, we get:

$\Rightarrow \int x^n~dx=\frac{x^{(n+1)}}{n+1}+C$

Where C is the integral constant.

The integration of a square function is also derived from this rule, i.e. $\int x^2~dx$. This rule can also be applied to other exponents. Let’s discuss with an example.

Example: Integrate $\int x^4~dx$.

$\int x^4~dx=\frac{x^{(4+1)}}{4+1}+C=\frac{x^5}{5}+C$

Where C is the integral constant.

Sum Rule

According to the sum rule, the integration of the sum of two functions equals the sum of their individual integrals.

$\int(f+g)dx=\int fdx+\int gdx$

Example: $\int (x+x^3)dx$

$=\int xdx +\int x^3 dx$

$=\frac{x^2}{2}+C+\frac{x^4}{4}+D$

Where C and D are the integration constants. So the integration can be written as:

$=\frac{x^2}{2} +\frac{x^4}{4}+K$

Now K is the integration constant here.

Difference Rule

The integration difference rule is identical to the sum rule.

$\int(f-g)dx=\int fdx-\int gdx$

Example: $\int (x-x^3)dx$

$=\int xdx -\int x^3 dx$

$=\frac{x^2}{2}+C-\frac{x^4}{4}+D$

Where C and D are the integration constants. So the integration can be written as:

$=\frac{x^2}{2} -\frac{x^4}{4}+K$

Now K is the integration constant here.

Multiplication by Constant

When a function is multiplied by a constant, the integration of that function is as follows:

$\Rightarrow \int cf(x)dx=c \int f(x)dx$

Example: $\int 4x~dx$

$=4\int x~dx$

$=4 \frac{x^2}{2} + C$

$=2x^2+C$

Where C is the integral constant.

Methods of Integration

In some cases, visual inspection is insufficient to determine the integral of a function. There are other ways to obtain the integral of a function that is reduced to its standard form. Other methods are mentioned below:

• Decomposition method

• Integration by Substitution

• Integration using Partial Fractions

• Integration by Parts

Decomposition Method

In this method we decompose the complicated function in the form of sum or difference of simple functions. For example, 2sinA.cosB can be decomposed as $\sin(A+B)-\sin(A-B)$.

Integration by Substitution

This method is known as reverse chain rule method and u-substitution. In this method we transform a complicated integral to a standard form using substitution.

For example, $\int 2x\cos(x^2+5)dx$

So, here we substitute $x^2+5=u$ then,

$\int \cos u ~du=\sin u+C$

Re-substitute, $x^2+5=u$

$\int 2x\cos(x^2+5)dx=\sin(x^2+5)+C$

Where C is integral constant.

Integration using Partial Fractions

If a rational function has a complicated denominator then this method will be used to decompose and integrate the function.

For example,$\dfrac{px+q}{(x-a)(x-b)}$ can be decomposed as $\frac{A}{x-a}+\frac{B}{x-b}$.

Integration by Parts

The product rule of integration is another name for this rule. When two functions are multiplied together, then this integration method is used.

The formula for integration by parts is:

$\int uv=u\int vdx-\int u’ (\int vdx)dx$

Here,

u is a function of u(x)

v is a function of v(x)

u’ will be the derivative of the function u(x)

Here we select u(first function) and v(second function) according to the ILATE rule.

• I - Inverse Functions

• L - Logarithm Functions

• A - Algebraic Functions

• T - Trigonometric Functions

• E - Exponential Functions

Did You know?

If the derivative of a function is given and asked to find the integration of it then we can simply write the function as answer. For example, 2x is derivative of $x^2$ and we want to integrate 2x then we can directly write $x^2$.

Solved Examples

1. $\int(x^4+4x^3+3x^2+2)dx$

Ans: $\int(x^4+4x^3+3x^2+2)dx$

$=\frac{x^5}{5}+4 \frac{x^4}{4}+3 \frac{x^3}{3}+2x+c$

$=\frac{x^5}{5}+x^4+x^3+2x+c$

Where c is integral constant.

2. $\int(3x^2+2x+4)\cos(x^3+x^2+4x)dx$

Ans: $\int(3x^2+2x+4)\cos(x^3+x^2+4x)dx$

Here we substitute $u=x^3+x^2+4x$

Therefore $du=(3x^2+2x+4)dx$

And the above equation becomes

$=\int\cos udu$

$=\sin u +c$

Putting the value of u in the equation we get

$=\sin(x^3+x^2+4x)+c$

Where c is the integral constant.

Practice Questions

Integrate the following functions:

1. $\int \frac{2x}{\sqrt(x^2-4)}dx$

2. $\int\frac{\ln (x)}{x^3}dx$

Conclusion

Integration is the process of determining the antiderivatives of functions, often called integrals. There are numerous approaches to integrating a function. The basic integration formulas are used to find the antiderivatives for a few standard integrals. There are several methods to use, including substitution, by parts integration, and partial fractions integration.