What is integration?

Integration is the process of finding the area of the region under the curve. It is done by drawing small rectangles covering up the area and then summing up these areas. The sum approaches a limit that is equal to the region under the curve of a function.

If the derivative of a function $F(x)$ is $f(x)$, then the integral of $f(x)$ is $F(x) + C$. It called as indefinite integrals. Where $C$ is the arbitrary constant and the antiderivative of $f(x)$ can be obtained by assigning a specific value to $C$.

Let $f(x) = x^3$ be a function.

• Derivative of $f(x)$ is $f'(x) = 3x^2$.

• Antiderivative of $3x^2$ is $f(x) = x^3$.

Integration: an inverse process of differentiation

Integration is the inverse process of differentiation. If the derivative of the function is given, we can find the integral by finding the original function.

Rules of integration

There are specific rules for finding the integrals of the functions. Some of the important rules are given below.

• Sum and difference rule

  • $\int [f(x)+g(x)] dx = \int f(x) dx + \int g(x) dx$

  • $\int [f(x)-g(x)] dx = \int f(x) dx - \int g(x) dx$

• Power rule

  • $\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$

• Exponential rule

  • $\int e^x dx = e^x + C$

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

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

• Constant multiplication rule

  • $\int a \; dx = ax + C, \quad \text{where a is the constant.}$

• Reciprocal rule

  • $\int \frac{1}{x} dx = \ln(x)+ C$

Methods of integration

There are some important methods that can be used to reduce the function to find its integral.

Decomposition method

The functions can be decomposed into a sum or a difference of functions (where individual integrals are known).

• If we want to integrate $\frac{x^2 - x + 1}{x^3} \,dx$, we decompose the function as follows.

Integration by substitution

This method allows us to change the variable of integration so that the integrand can be integrated easily.

Example

• Let's find the integral of $f(x) = \sin(mx)$ using substitution.

• Let $mx = t$

• So, $m \frac{dx}{dt} = 1$

Note:

The substitution method can also use trigonometric identities. Some are given below.

• $\int tan x \; dx = \log|\sec x| + C$

• $\int cot x \; dx = \log|\sin x| + C$

• $\int cosec x \; dx = \log|\cosec x - \cot x| + C$

• $\int sec x \; dx = \log|\sec x + \tan x| + C$

Integration using partial fraction

Suppose we want to find $y = \int \frac{P(x)}{Q(x)} dx$ where $\frac{P(x)}{Q(x)}$ is an improper rational function.

We can reduce it as follows.

Where,

• $T(x)$ is a polynomial in $x$

• $\frac{P_{1}(x)}{Q(x)}$ is a proper rational function.

Give below is a table which shows some common rational functions and their respective partial functions.

Example

Let's find the integral of $f(x) = \frac{1}{(x+1)(x+2)}$.

Using partial fraction,

On comparing the above equation (A), we get:

From this above form, we have two equations as given below.

Solving the above equations give us $A = 1$ and $B = -1$.

Thus, equation A can be rewritten as follows.

Now, the solution of the integral is given below.

Integration by parts

This method is normally used to find the integral of two functions.

Using the product rule of derivatives, we have $\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$.

Integration on both sides of the equation gives us:

Or,

Example

Let's find the integral of $xe^{x}$ using integration by parts.

We get the following results:

• $\int e^{ax} \sin bx \,dx = \frac{e^{ax}}{a^2 + b^2} \left[ a \sin bx - b \cos bx \right] + C$

• $\int e^{ax} \cos bx \,dx = \frac{e^{ax}}{a^2 + b^2} \left[ a \cos bx + b \sin bx \right] + C$

Important notes

Note:

• Integration is an inverse process of differentiation.

• Do not forget to add the constant of integration after determining the integral of the function.

• If two functions, say $f(x)$ and $g(x)$ have the same derivatives, then $|f(x)-g(x)|= C$, where $C$ is a constant.

Read more

• What is differentiation?