Integration: definition, formulas, properties, and examples

Basic concepts of Integration:

In Differential Calculus, we have learned to find the derivative/differential of a differentiable function. A natural question that may come to one’s mind is that what is the inverse method of differential calculus. More precisely, if we know the derivative of a function, then can we determine the function? Let’s understand this with an example.

Let $f(x)$ be a function with differential $4x^3$, that is $f'(x)$ $=\frac{d}{dx}(f(x))$ $=4x^3$. From this information, can we determine the function $f(x)$? The answer is Yes. The process to find $f(x)$ is the main purpose to study Integral Calculus

In the above example, given that $\frac{d}{dx}(f(x))=4x^3$. Observe that $f(x)=x^4$ satisfies this condition. The function $f(x)=x^4$ is obtained by integrating $4x^3$. On the other hand. we can get back the function $f(x)=4x^3$ by differentiating $x^4$. Thus we can say that:

Integration is called the inverse method of differentiation.

What is Integration?

1. The method to find a function from its known differential is called integration.

2. The function we determine from the differential of a given function is called integral.

Definition and Notation of Integration:

Let $f(x)$ be a function of $x$ such that the derivative of another function $F(x)$ is equal to $f(x)$, that is, $\frac{d}{dx}(F(x))=f(x)$. Then $F(x)$ is called an Indefinite Integral (or Primitive Function or Anti-derivative) of $f(x)$ with respect to $x$.

In symbolic notation, we write $\int f(x) \, dx=F(x)$ [Read as: the integral of $f(x) \, dx$ is $F(x)$.] The symbol $\int$ is called the sign of integration. As the process of integration is considered one of the methods of finding sums, the symbol $\int$ comes from the first letter of the word “Sum”.

Summary:

$\frac{d}{dx}(F(x))=f(x)$ and $\int f(x) \, dx =F(x)$.

In other words, $\int \frac{d}{dx}(F(x)) \, dx =F(x)$.

Let’s understand the above facts with examples.

Example 1. Recall that $\frac{d}{dx}(\sin x)=\cos x$.

So by the definition of integration, $\int \cos x \, dx=\sin x$

Example 2. We know that $\frac{d}{dx}(e^x)=e^x$.

Thus by definition, $\int e^x \, dx=e^x$.

Application of Integration:

The method of integration is generally used to find the area of a region bounded by curves.

List of all Integration Formulas

It is helpful to keep all integral formulas handy while solving problems of integrations. Here we list all integral formulas in one place.

Basic Integration Formulas:

1.  $\int dx=x+c$

2.  $\int x^n \, dx=\frac{x^{n+1}}{n+1}+c; \, (n \neq -1)$

3.  $\int e^x \, dx=e^x+c$

4.  $\int a^x \, dx=\frac{a^x}{\log|a|}+c;$ $\, (a>0, a \neq 1)$

5.  $\int \frac{1}{x} \, dx=\log x+c$

6.  $\int \frac{dx}{\sqrt{1-x^2}}=\sin^{-1} x+c$

7.  $\int \frac{dx}{1+x^2}=\tan^{-1} x+c$

Trigonometric Functions Integration Formulas:

1.  $\int \sin x \, dx=-\cos x+c$

2.  $\int \cos x \, dx=\sin x+c$

3.  $\int \sec x \, dx$ $=\log |\sec x + \tan x|+c$

4.  $\int \text{cosec} \, x \, dx$ $=\log |\text{cosec}\, x – \cot x|+c$

5.  $\int \tan x \, dx=\log |\sec x|+c$

6.  $\int \cot x \, dx=\log |\sin x|+c$

7.  $\int \sec^2 x \, dx=\tan x+c$

8.  $\int \text{cosec}^2 \, x \, dx=-\cot x+c$

9.  $\int \sec x \tan x \, dx=\sec x+c$

10.  $\int \text{cosec}\, x \cot x \, dx$ $=-\text{cosec}\, x+c$

Integration of mod x

Integration of √x

Integration Formulas (Substitution Method):

1. $\int e^{mx} \, dx=\frac{e^{mx}}{m}+c$

2. $\int a^{mx} \, dx=\frac{a^{mx}}{m\log|a|}+c;$ $\,(a>0, a \neq 1)$

3.  $\int \sin mx \, dx=-\frac{\cos mx}{m}+c$

4.  $\int \cos mx \, dx=\frac{\sin mx}{m}+c$

5.  $\int \sec^2 mx \, dx=\frac{\tan mx}{m}+c$

6.  $\int \text{cosec}^2 \, mx \ dx$ $=-\frac{\cot mx}{m}+c$

7.  $\int \sec mx \tan mx \, dx$ $=\frac{\sec mx}{m}+c$

8.  $\int \text{cosec}\, mx \cot mx \, dx$ $=-\frac{\text{cosec}\, mx}{m}+c$

Some Special Integration Formulas:

1.  $\int \frac{dx}{x^2+a^2}=\frac{1}{a}\tan^{-1}\frac{x}{a}+c$

2.  $\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\log |\frac{x-a}{x+a}|+c$

3.  $\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\log |\frac{a-x}{a+x}|+c$

4.  $\int \frac{dx}{\sqrt{x^2+a^2}}$ $=\log |x+\sqrt{x^2+a^2}|+c$

5.  $\int \frac{dx}{\sqrt{x^2-a^2}}$ $=\log |x-\sqrt{x^2-a^2}|+c$

6.  $\int \frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac{x}{a}+c$

7.  $\int \frac{dx}{x\sqrt{x^2-a^2}}=\frac{1}{a}\sec^{-1}\frac{x}{a}+c$

8.  $\int \sqrt{x^2+a^2}\, dx=\frac{x\sqrt{x^2+a^2}}{2}$ $+\frac{a^2}{2}\log |x+\sqrt{x^2+a^2}|+c$

9.  $\int \sqrt{x^2-a^2}\, dx=\frac{x\sqrt{x^2-a^2}}{2}$ $-\frac{a^2}{2}\log |x+\sqrt{x^2-a^2}|+c$

10.  $\int \sqrt{a^2-x^2}\, dx=\frac{x\sqrt{a^2-x^2}}{2}$ $+\frac{a^2}{2}\sin^{-1}\frac{x}{a}+c$

Integration by Parts Formulas

To find the integration of the product of functions, we use the technique of integration by parts. If $u$ and $v$ are two functions of $x$, then the integration formula of their product, that is, the integration of $u v \, dx$ is given as follows:

$\int u v \, dx$ $=u\int v \, dx -\int [\frac{du}{dx}( \int v \, dx )]\, dx$

The above formula is called the integration formula by parts. In this formula, we call the function $u$ as the first function and $v$ as the second function. If we have to find the integration of the product of functions, then we have to choose the first or second function according to the rule below (LIATE rule):

L -> logarithmic functions (example $\log x$)
I ->  inverse functions (example $\sin^{-1}x$)
A -> algebraic functions (example x2)
T -> trigonometric functions (example $\sin x$)
E -> exponential functions (example ex)

For example, if we have to evaluate $\int x \sin x\, dx$ using integration by parts formula, then according to the LIATE rule, we have to take $x$ as the first function and $\sin x$ as the second function. So applying the above rule of integration of parts with $u=x$ and $v=\sin x$, we have

$\int x \sin x \, dx$

$=x\int \sin x \, dx$  $- \int [\frac{dx}{dx}( \int \sin x \, dx )]\, dx$

$=-x\cos x -\int[1 \cdot (-\cos x)] dx$  $+c$

$=-x\cos x+\int \cos x \, dx+c$

$=-x\cos x+\sin x+c$

Some applications of the above formula

(i)  $\int e^x[f(x)+f'(x)] \, dx$ $=e^x f(x)+c$

(ii)  $\int e^{ax} \sin bx \, dx$ $=\frac{e^{ax}(a\sin bx – b \cos bx)}{a^2+b^2}$$+c (iii) \int e^{ax} \cos bx \, dx =\frac{e^{ax}(a\cos bx + b \sin bx)}{a^2+b^2}$$+c$

Properties of Definite Integrals

1. If $f(x)=\frac{d}{dx}(\phi(x))$ then we have

$\int_a^b f(x) \, dx=\phi(b)-\phi(a)$.

The above is the main property of definite integrals.

2. $\int_a^b f(x) \, dx=\int_a^b f(t) \, dt$

3. $\int_a^b f(x) \, dx=-\int_b^a f(x) \, dx$

4. $\int_a^b f(x) \, dx$ $=\int_a^c f(x) \, dx + \int_b^c f(x)\, dx$ where $a<c<b.$

5. $\int_a^a f(x) \, dx=0$

6. $\int_0^a f(x) \, dx=\int_0^a f(a-x) \, dx$

7. If $f(a+x)=f(x)$ then $\int_0^{na} f(x) \, dx=n \int_0^a f(x) \, dx$

8. $\int_0^{2a} f(x) \, dx= 2\int_0^a f(x) \, dx$ if $f(a-2x)=f(x)$

9. $\int_0^{2a} f(x) \, dx=0$ if $f(a-2x)=-f(x)$

10. $\int_{-a}^{a} f(x) \, dx=2\int_0^a f(x) \, dx$ if $f(-x)=f(x)$

11. $\int_{-a}^{a} f(x) \, dx=0$ if $f(-x)=-f(x)$

12. $\int_a^b f(x) \, dx$ $=\int_a^b(a+b-x) \, dx$

Few Examples

Now we will provide few examples with solutions to familiar the students with the above integral formulas.

Example 1: $\int 3x^2 \, dx$

$=3\int x^2 \, dx$

$=3 \cdot \frac{x^3}{3}+c$

$=x^3+c$
Example 2: $\int \frac{dx}{x^2+25}$

$=\int \frac{dx}{x^2+5^2}$

$=\frac{1}{5}\tan^{-1}\frac{x}{5}+c$
Example 3: $\int e^x(x+1) \, dx$

$=xe^x+c$

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