Indefinite and Definite Integration

Indefinite Integration

Integration means summation. Integration originated during the course of finding the area of a plane figure. Integration is the reverse of differentiation. It is also called as the antiderivative. In this section, aspirants will learn the list of important formulas, how to use integral properties to solve integration problems, integration methods and many more.

The process of finding an indefinite integral of a given function is called integration of the function. Integrating a function f(x) means finding a function $\Phi (x) \: such \: that \frac{d}{dx}\left ( \Phi (x) \right ) = f(x))$ .

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

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:

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

Standard integral form $f\left( ax+b \right)$ $\int{{{\left( ax+b \right)}^{n}}dx=\frac{{{\left( ax+b \right)}^{n+1}}}{\left( n+1 \right)a}}$ $\int{\frac{1}{ax+b}}=dx=\frac{\ell n\left| ax+b \right|}{a}+c$

and so on…

we follow as below

If $\int{f\left( x \right)}=F\left( x \right)+c$ then $\int{f\left( ax+b \right)}=\frac{F\left( ax+b \right)}{a}+c$

Methods of Integration

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

Substitution method

A new variable is introduced such that direct integration formula can be applied.

Problems on Indefinite Integration

Problem 1. $\int{2x\,\,\sin \left( {{x}^{2}} \right)dx}$

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

the $2x\,\,dx=dt$ $\int{\sin t\,\,dt=-\cos t+c}$ $=-\cos {{x}^{2}}+c$

Problem 2. $\int{{{\sin }^{3}}x{{\cos }^{5}}x\,\,dx}$

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 $\cos x=t$

∴ $\,\,\,-\sin x\,dx=dt$ $=\int{\left( 1-{{t}^{2}} \right)}{{t}^{5}}\left( -dt \right)$ $=\int{{{t}^{5}}-{{t}^{7}}}dt$ $=-\left( \frac{{{t}^{6}}}{6}-\frac{{{t}^{8}}}{8} \right)$ $=-\frac{{{\cos }^{6}}x}{6}+\frac{{{\cos }^{8}}x}{8}+c$

Note: 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. $\sqrt{\frac{x-a}{x-b}}\rightarrow{{}}x=a{{\sec }^{2}}\theta -b$

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. $\int{\sqrt{{{x}^{2}}-{{a}^{2}}}}dx=\frac{x}{2}\sqrt{{{x}^{2}}-{{a}^{2}}}+\frac{{{a}^{2}}}{2}\ell n\left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|$

Problem Type 1:

Problem Type 2:

here

Important forms

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

Problem Type 3:

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

⇒ $I=\int{\frac{1}{2}}\frac{{{\left( {{x}^{2}}+3x+4 \right)}^{1}}}{{{x}^{2}}+3x+4}-\frac{1}{2}\left( \frac{1}{{{x}^{2}}+3x+4} \right)$ $=\frac{1}{2}\log \left( {{x}^{2}}+3x+4 \right)-\frac{1}{2}\int{\frac{1}{{{\left( x+\frac{3}{2} \right)}^{2}}+\frac{7}{4}}}$ $=\frac{1}{2}\log \left( {{x}^{2}}+3x+4 \right)-\frac{1}{2}\frac{1}{\sqrt{\frac{7}{4}}}{{\tan }^{-1}}\frac{\left( x+\frac{3}{2} \right)}{\sqrt{\frac{7}{4}}}$ $=\frac{1}{2}\log \left| {{x}^{2}}+3x+4 \right|-\frac{1}{\sqrt{7}}{{\tan }^{-1}}\left( \frac{2x+3}{\sqrt{7}} \right)$

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

Problem Type 4:

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.

Put $x=1\Rightarrow A=\frac{1}{2}$ $x=-1\Rightarrow \,\,C=\frac{-1}{2}$ $B=\frac{3}{2}$ $I=\int{\frac{\frac{1}{2}}{\left( x-1 \right)}}dx+\frac{3}{2}\int{\frac{dx}{\left( x+1 \right)}}+\left( \frac{-1}{2} \right)\int{\frac{dx}{{{\left( x-2 \right)}^{2}}}}$ $=\frac{1}{2}\ell n\left| x-1 \right|+\frac{3}{2}\left( n\left| x+1 \right. \right)+\frac{1}{2\left( x+2 \right)}+c$ $\int{\frac{dx}{\left( x+2 \right)\left( {{x}^{2}}+1 \right)}}=\frac{A}{x+2}+\frac{Bx+C}{{{x}^{2}}+1}$

Comparing by $x=-2\Rightarrow \,\,A=\frac{1}{5}$ by ${{x}^{2}}$ coeff. $B=\frac{-1}{2}$ $C=\frac{2}{5}$ $I=\frac{1}{5}\left| \frac{dx}{x+2} \right.-\frac{1}{5}\int{\frac{x}{{{x}^{2}}+1}}\,dx$ $I=\frac{1}{5}\ell n\left| x+2 \right|-\frac{1}{10}\ell n\left( {{x}^{2}}+1 \right)+\frac{2}{5}{{\tan }^{-1}}\left( x \right)$

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 II

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 … $a+ar+a{{r}^{2}}….a{{r}^{n-1}}=\frac{a\left( 1-{{r}^{n}} \right)}{1-r}$ $\sin \alpha +\sin \left( \alpha +\beta \right)+\sin \left( \alpha +2\beta \right)….\sin \left( \alpha +\left( n-1 \right)\beta \right)$ $=\frac{\sin \left( n\frac{\beta }{2} \right)}{\sin \left( \frac{\beta }{2} \right)}\sin \left[ \frac{\alpha +\alpha +\left( n-1 \right)\beta }{2} \right]$ $=\frac{\sin \left( n\frac{diff}{2} \right)}{\sin \left( \frac{diff}{2} \right)}\sin \left( \frac{1st+last}{2} \right)$ $=\frac{\sin \left( \frac{n\,diff}{2} \right)}{\sin \left( \frac{diff}{2} \right)}\cos \left( \frac{1st+last}{2} \right)$ $1-\frac{1}{{{2}^{2}}}+\frac{1}{{{3}^{2}}}-\frac{1}{{{4}^{2}}}+\frac{1}{{{5}^{2}}}-\frac{1}{{{6}^{2}}}=\frac{{{\pi }^{2}}}{12}$ $1+\frac{1}{{{2}^{2}}}+\frac{1}{{{3}^{2}}}+\frac{1}{{{4}^{2}}}+\frac{1}{{{5}^{2}}}….=\frac{{{\pi }^{2}}}{12}$ $\int\limits_{a}^{b}{{{e}^{x}}dx=\underset{n\to \infty }{\mathop{\lim }}\,}h\left[ {{e}^{a}}+{{e}^{a+h}}+{{e}^{a+2h}}…{{e}^{a+\left( n-1 \right)h}} \right]$ $=h{{e}^{a}}\left[ 1+{{e}^{n}}+{{e}^{2h}}+…. \right]$ $={{e}^{a}}\left[ {{e}^{nh}}-1 \right]\frac{h}{\left( {{e}^{n}}-1 \right)}$ $={{e}^{b}}-{{e}^{a}}$ $\int\limits_{a}^{b}{\sin x\,dx=\underset{n\to \infty }{\mathop{\lim }}\,}h\left[ \sin a+\sin \left( a+n \right)+…\sin \left( a+\left( n-1 \right)h \right) \right]$