Indefinite and Definite Integration

In Calculus, integration and differentiation are two most important concepts. Integration originated during the course of finding the area of a plane figure whereas differentiation is a process of finding a function that outputs the rate of change of one variable with respect to another variable. Integration is the reverse of differentiation. It is also called as the antiderivative. In this section, aspirants will learn about the indefinite and definite Integration list of important formulas, how to use integral properties to solve integration problems, integration methods and many more.

Indefinite Integration

The indefinite integral is defined as a function that will describe an area under the function’s curve from an undefined point to another arbitrary point. Read more.

Standard formulas for Indefinite Integration

1. $\int{{{x}^{n}}dx=\frac{{{x}^{n+1}}}{n+1}}+c$
2. $\int{{{a}^{x}}dx}=\frac{{{a}^{x}}}{\ell n\,a}+c$
3. $\int{{{e}^{x}}}dx={{e}^{x}}+c$
4. $\int{\frac{1}{x}}dx=\ell n\,x+c$
5. $\int{\sin x\,dx=-\cos x+c}$
6. $\int{\cos x\,dx=\sin x+c}$
7. $\int{{{\sec }^{2}}x\,\,dx=\tan x+c}$
8. $\int{\cos e{{c}^{2}}x\,\,dx=-\cot x\,+c}$
9. $\int{\sec x\tan xdx=\sec x+c}$
10. $\int{\cos ec\,\,x\,\,}\cot x\,dx=-\cos ec\,x+c$
11. $\int{\frac{1}{\sqrt{1-{{x}^{2}}}}dx\begin{matrix} = \\ = \\ \end{matrix}}\left\{ \begin{matrix} {{\sin }^{-1}}x+c \\ -{{\cos }^{-1}}x+c \\ \end{matrix} \right.$
12. $\int{\frac{1}{1+{{x}^{2}}}}dx\begin{matrix} = \\ = \\ \end{matrix}\left\{ \begin{matrix} {{\tan }^{-1}}x+c \\ -{{\cot }^{-1}}x+c \\ \end{matrix} \right.$.
13. $\frac{1}{x\sqrt{{{x}^{2}}-1}}=\left\{ \begin{matrix} {{\sec }^{-1}}x+c \\ -\cos e{{c}^{-1}}x+c \\ \end{matrix} \right.$

⇒ Also Read – Introduction to Integration

Methods of Integration

1. Substitution Method
2. Integration by partial fractions
3. By parts
4. Euler substitution
5. Reduction method

Properties of Indefinite Integration

1. $k\int{f\left( x \right)dx=k\int{f\left( x \right)}}$
2. $\int{\left( f\left( x \right)+g\left( x \right) \right)dx=\int{f\left( x \right)dx+\int{g\left( x \right)dx}}}$

Below are some Illustrations based on integration properties and substitution method:

Illustration 1: Solve $\int{\frac{{{\left( 1+x \right)}^{3}}}{\sqrt[3]{{{x}^{2}}}}}dx$

Solution:  $\int{\frac{{{\left( 1+x \right)}^{3}}}{\sqrt[3]{{{x}^{2}}}}}dx=\int{\frac{1+3x+3{{x}^{2}}+{{x}^{3}}}{{{x}^{2/3}}}}dx$ $=\int({{{x}^{-2/3}}+3{{x}^{1/3}}+3{{x}^{4/3}}}+{{x}^{7/3}})dx$ $=3{{x}^{1/3}}+\frac{9}{4}{{x}^{4/3}}+\frac{9}{7}{{x}^{7/3}}+\frac{3}{10}{{x}^{10/3}}+c$

Illustration 2: Solve  $\int{{\sec }^{2}}x\cos e{{c}^{2}}x\,\,dx$

Solution: $\int{{{\sec }^{2}}x\cos e{{c}^{2}}x\,\,dx=\int{\frac{{{\sin }^{2}}x+{{\cos }^{2}}x}{{{\sin }^{2}}x\,{{\cos }^{2}}x}}dx}$ $=\int{{{\sec }^{2}}x+\cos e{{c}^{2}}x\,\,dx}$ $=\tan x-\cot x+c$

Illustration 3. $\int{2x\,\,\sin \left( {{x}^{2}} \right)dx}$

Solution: Put ${{x}^{2}}=t$

then, $2x\,\,dx=dt$,

Illustration 4. $\int{{{\sin }^{3}}x{{\cos }^{5}}x\,\,dx}$

Solution: $\int{{{\sin }^{2}}x{{\cos }^{5}}x\sin x\,\,dx}$ $=\int{\left( 1-{{\cos }^{2}}x \right)}{{\cos }^{5}}x\,\,\sin x\,\,dx$

put $\cos x=t$

∴ $\,\,\,-\sin x\,dx=dt$

so, new equation is: $\int{\left( 1-{{t}^{2}} \right)}{{t}^{5}}\left( -dt \right)$ $=\int{{{t}^{5}}-{{t}^{7}}}dt$ $=-\left( \frac{{{t}^{6}}}{6}-\frac{{{t}^{8}}}{8} \right)$

Resubstituting the value, we get

Important formulae set for Indefinite Integration

1. $\int{\frac{dx}{\sqrt{{{a}^{2}}-{{x}^{2}}}}}={{\sin }^{-1}}\left( \frac{x}{a} \right)+c$
2. $\int{\frac{dx}{\sqrt{{{x}^{2}}-{{a}^{2}}}}=\ell n\left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|}$
3. $\int{\frac{dx}{\sqrt{{{x}^{2}}+{{a}^{2}}}}=\ell n\left| x+\sqrt{{{x}^{2}}+{{a}^{2}}} \right|}$
4. $\int{\frac{1}{{{a}^{2}}+{{x}^{2}}}}dx=\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right).$
5. $\int{\frac{1}{{{a}^{2}}-{{x}^{2}}}}dx=\frac{1}{2a}\ell n\left| \frac{a+x}{a-x} \right|.$
6. $\int{\frac{1}{{{x}^{2}}-{{a}^{2}}}}dx=\frac{1}{2a}\ell n\left| \frac{x-a}{x+a} \right|.$
7. $\int{\sqrt{{{a}^{2}}-{{x}^{2}}}}dx=\frac{x}{2}\sqrt{{{a}^{2}}-{{x}^{2}}}+\frac{{{a}^{2}}}{2}{{\sin }^{-1}}\left( \frac{x}{a} \right).$
8. $\int{\sqrt{{{a}^{2}}+{{x}^{2}}}}dx=\frac{x}{2}\sqrt{{{x}^{2}}+{{a}^{2}}}+\frac{{{a}^{2}}}{2}\ell n\left| x+\sqrt{{{x}^{2}}+{{a}^{2}}} \right|$
9. $\int{\sqrt{{{x}^{2}}-{{a}^{2}}}}dx=\frac{x}{2}\sqrt{{{x}^{2}}-{{a}^{2}}}+\frac{{{a}^{2}}}{2}\ell n\left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|$

 Types of substitutions:

Type I

Rule:

1. If m, n one of them odd then substitute for even power.
2. If both odd then substitute either of them.
3. If both even use trigonometric identities.

Type II: Substitution of Trigonometric functions.

1. $\sqrt{{{a}^{2}}-{{x}^{2}}}\rightarrow{{}}x=a\sin \theta$
2. $\sqrt{{{x}^{2}}+{{a}^{2}}}\rightarrow{{}}x=a\tan \theta$
3. $\sqrt{{{x}^{2}}-{{a}^{2}}}\rightarrow{{}}x=a\sec \theta$
4. $\sqrt{\frac{x-a}{b-x}}\rightarrow{{}}x=a{{\cos }^{2}}\theta +b{{\sin }^{2}}\theta$
5. $\sqrt{\frac{x-a}{x-b}}\rightarrow{{}}x=a{{\sec }^{2}}\theta -b$

Type III:

We express $px+q$ as $m{{\left( a{{x}^{2}}+bx+c \right)}^{1}}+n$ then this gets changed to standard integral.

Biquadratic Substitutions

1. $I=\int{f\left( x+\frac{1}{x} \right)}\left( 1-\frac{1}{{{x}^{2}}} \right)dx$

Put $x+\frac{1}{x}=t$

2. $I=\int{f\left( x-\frac{1}{x} \right)}\left( 1+\frac{1}{{{x}^{2}}} \right)dx$

Put $\left( x-\frac{1}{x} \right)=t$

Solved Problems

Problem 1: Solve $\int{\frac{dx}{\sqrt{{{x}^{2}}+2x+2}}}$

Solution: $\int{\frac{dx}{\sqrt{{{x}^{2}}+2x+2}}}=\int{\frac{dx}{\sqrt{{{\left( x+1 \right)}^{2}}+1}}}$ $=\log \left| \left( x+1 \right)+\sqrt{{{\left( x+1 \right)}^{2}}+1} \right|.$

Problem 2: Solve $\int{\sqrt{5{{x}^{2}}+2x+3}}\,\,dx$

Solution: $\int{\sqrt{5{{x}^{2}}+2x+3}}\,\,dx$

here

Problem 3: Evaluate $I=\int{\frac{x+1}{{{x}^{2}}+3x+4}}dx$

Solution: $I=\int{\frac{x+1}{{{x}^{2}}+3x+4}}dx$,

∴ $a=\frac{1}{2}\,\,\,\,b=\frac{-1}{2}$

⇒ $I=\int{\frac{1}{2}}\frac{{{\left( {{x}^{2}}+3x+4 \right)}^{1}}}{{{x}^{2}}+3x+4}-\frac{1}{2}\left( \frac{1}{{{x}^{2}}+3x+4} \right)$,

Problem 4: Solve $\int{\frac{1+{{x}^{2}}}{1+{{x}^{4}}}}\,dx$

Solution: $\int{\frac{1+{{x}^{2}}}{1+{{x}^{4}}}}\,dx=\int{\frac{\left( 1+\frac{1}{{{x}^{2}}} \right)}{\left( \frac{1}{{{x}^{2}}}+{{x}^{2}} \right)}}\,dx$,

Integration by Partial Fraction

Integrals of rational functions can be evaluated by splitting them into partial fractions.

If the degree of $f<$, the degree of g it is called a proper fraction.

Otherwise, it is an improper fraction.

Then $\frac{f\left( x \right)}{g\left( x \right)}=h\left( x \right)+\frac{d\left( x \right)}{g\left( x \right)}$ , degree of d < degree of g.

Cases:

1. When $g\left( x \right)$ is expressed as product of non-repeating linear factors.

2. Some factors of g are repeating then $g\left( x \right)={{\left( x-a \right)}^{k}}\left( x-{{a}_{1}} \right)\left( x-{{a}_{2}} \right)…$ $\frac{f}{g}=\frac{{{A}_{1}}}{\left( x-a \right)}+\frac{{{A}_{2}}}{{{\left( x-a \right)}^{2}}}+…..\frac{{{A}_{k}}}{{{\left( x-a \right)}^{n}}}+…$

3. $g\left( x \right)$ has a quadratic term then $\frac{f}{g}=\frac{Ax+B}{a{{x}^{2}}+bx+c}$

Where A & B are constants determined by comparing coefficients.

Let us understand with the help of examples:

Example 1: Solve $\int{\frac{dx}{\left( x+1 \right)\left( x-2 \right)}}$

Solution: $\int{\frac{dx}{\left( x+1 \right)\left( x-2 \right)}}=\int{\frac{a}{x+1}+\frac{b}{x-2}}\,dx$,

Example 2: Solve $\int{\frac{2{{x}^{2}}-1}{\left( x-1 \right){{\left( x+1 \right)}^{2}}}}dx$

Solution: $\int{\frac{2{{x}^{2}}-1}{\left( x-1 \right){{\left( x+1 \right)}^{2}}}dx=\int{\frac{A}{x-1}+\frac{B}{\left( x+1 \right)}}+\frac{C}{{{\left( x+1 \right)}^{2}}}}$,

Put $x=1\Rightarrow A=\frac{1}{2}$,

Example 3: Solve $\int{\frac{dx}{\left( x+2 \right)\left( {{x}^{2}}+1 \right)}}$

Solution: $\int{\frac{dx}{\left( x+2 \right)\left( {{x}^{2}}+1 \right)}}=\frac{A}{x+2}+\frac{Bx+C}{{{x}^{2}}+1}$

Comparing by $x=-2\Rightarrow \,\,A=\frac{1}{5}$ and by ${{x}^{2}}$ coeff. $B=\frac{-1}{2}$ and $C=\frac{2}{5}$,

on substituting the values and integrating we have

Integration of Trigonometric Functions

Type 1: $I=\int{{{\sin }^{m}}x{{\cos }^{n}}xdx}$

1. If m –odd put $\cos x=t$

2. If n odd put $\sin x=t$

3. If m, n rationales then put $\tan x=t$

4. If both even then use reduction method

Let us understand with the help of an example:

Where $t=\sin x$ $=\int{{{t}^{-6}}-{{t}^{-4}}dt}$ $=-\frac{1}{5si{{n}^{5}}x}+\frac{1}{3{{\sin }^{3}}x}+c$

Type 2: $\int{\frac{dx}{a\cos x+b\sin x+c}}$

Put $t=\tan \left( \frac{x}{2} \right)$

Illustration:

Type 3

Rule:

Divide both numerator & denominator by ${{\cos }^{2}}x$

Illustration:

Definite Integrals

Choose n divide [a, b] in n parts of width $h=\frac{b-a}{n}=$. partition of interval putting (n – 1) parts in between

Definition of Definite Integral as limit of a solution

Integrals using limits

Evaluate as Limit of a Sum

GP … $a+ar+a{{r}^{2}}….a{{r}^{n-1}}=\frac{a\left( 1-{{r}^{n}} \right)}{1-r}$