4.3 STANDARD INTEGRALS

\[Integrals\ of\ the\ form\ \int \frac{dx}{a^2 \pm x^2}, \int \frac{dx}{x^2-a^2}\ ,\ \int \frac{dx}{\sqrt{a^2 – x^2}}\]
\[\int {\sqrt{a^2 - x^2}}\ dx\ and\ \int {\sqrt{x^2 \pm a^2}}\ dx\]

List of  Formulae:

\[1.\ \int \frac{dx}{a^2 + x^2} = \frac{1}{a}\ {tan}^{-1}(\frac{x}{a}) +c\ (or)\ \int \frac{dx}{x^2 + a^2} = \frac{1}{a}\ {tan}^{-1} (\frac{x}{a}) +c\]
\[2.\ \int \frac{dx}{a^2 - x^2} = \frac{1}{2a}\ log\ (\frac{a + x}{a - x}) + c\]
\[3.\ \int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\ log\ (\frac{x - a}{x + a}) + c\]
\[4.\ \int \frac{dx}{{\sqrt{a^2 - x^2}}} = {sin}^{-1}(\frac{x}{a})\ + c\]
\[5.\ \int {\sqrt{a^2 - x^2}}\ dx = \frac{x}{2}\ {\sqrt{a^2 - x^2}}\ + \frac{a^2}{2} {sin}^{-1}(\frac{x}{a})\ + c\]
\[6.\ \int {\sqrt{x^2 + a^2}}\ dx = \frac{x}{2}\ {\sqrt{x^2 + a^2}}\ + \frac{a^2}{2} {sin\ h}^{-1}(\frac{x}{a})\ + c\]
\[7.\ \int {\sqrt{x^2 - a^2}}\ dx = \frac{x}{2}\ {\sqrt{x^2 - a^2}}\ - \frac{a^2}{2} {cos\ h}^{-1}(\frac{x}{a})\ + c\]

Example :

\[Evaluate:\ \int \frac{dx}{x^2 + 16}\]

Soln:

\[W.K.T\ \int \frac{dx}{x^2 + a^2} = \frac{1}{a}\ {tan}^{-1} (\frac{x}{a}) +c\]
\[\int \frac{dx}{x^2 + 16} = \int \frac{dx}{x^2 + 4^2}\]
\[ = \frac{1}{4}\ {tan}^{-1} (\frac{x}{4}) +c\]

Example :

\[Evaluate:\ \int \frac{dx}{4x^2 + 9}\]

Soln:

\[\int \frac{dx}{4x^2 + 9} = \frac{1}{4}\ \int \frac{dx}{x^2 + \frac{9}{4}} = \frac{1}{4}\ \int \frac{dx}{x^2 + (\frac{3}{2})^2}\]
\[W.K.T\ \int \frac{dx}{x^2 + a^2} = \frac{1}{a}\ {tan}^{-1} (\frac{x}{a}) +c\]
\[\int \frac{dx}{4x^2 + 9} = \frac{1}{4}\ × \frac{2}{3}\ {tan}^{-1} (\frac{x}{\frac{3}{2}}) +c\]
\[ = \frac{1}{6}\ {tan}^{-1} (\frac{2}{3}) +c\]

Example :

\[Evaluate:\ \int \frac{dx}{4 + 9 x^2}\]

Soln:

\[\int \frac{dx}{4 + 9 x^2} = \int \frac{dx}{9 x^2+ 4} = \frac{1}{9}\ \int \frac{dx}{x^2 + \frac{4}{9}} = \frac{1}{9}\ \int \frac{dx}{x^2 + (\frac{2}{3})^2}\]
\[W.K.T\ \int \frac{dx}{x^2 + a^2} = \frac{1}{a}\ {tan}^{-1} (\frac{x}{a}) +c\]
\[\int \frac{dx}{4 + 9 x^2} = \frac{1}{9}\ × \frac{3}{2}\ {tan}^{-1} (\frac{x}{\frac{2}{3}}) +c\]
\[ = \frac{1}{6}\ {tan}^{-1} (\frac{3x}{2}) +c\]

Example :

\[Evaluate:\ \int \frac{dx}{(3x + 2)^2 + 16}\]

Soln:

Put      u  = 3x + 2

\[\frac{du}{dx}= \ 3\]
\[dx = \frac{1}{3}\ du\]
\[\int \frac{dx}{(3x + 2)^2 + 16} = \frac{1}{3}\ \int \frac{du}{u^2 + 4^2}\]
\[W.K.T\ \int \frac{dx}{x^2 + a^2} = \frac{1}{a}\ {tan}^{-1} (\frac{x}{a}) +c\]
\[\frac{1}{3}\ \int \frac{du}{u^2 + 4^2} = \frac{1}{3}\ × \frac{1}{4}\ {tan}^{-1} (\frac{u}{4}) +c\]
\[ = \frac{1}{12}\ {tan}^{-1} (\frac{3x + 2}{4}) +c\]
\[\int \frac{dx}{(3x + 2)^2 + 16} = \frac{1}{12}\ {tan}^{-1} (\frac{3x + 2}{4}) +c\]

Example :

\[Evaluate:\ \int \frac{dx}{9 - (3x - 2)^2}\]

Soln:

Put      u  = 3x - 2

\[\frac{du}{dx}= \ 3\]
\[dx = \frac{1}{3}\ du\]
\[\int \frac{dx}{9 - (3x - 2)^2}= \frac{1}{3}\ \int \frac{du}{3^2 - u^2}\]
\[W.K.T\ \int \frac{dx}{a^2 - x^2} = \frac{1}{2a}\ log\ (\frac{a + x}{a - x}) + c\]
\[\frac{1}{3}\ \int \frac{du}{3^2 - u^2} = \frac{1}{3}\ [ \frac{1}{2 × 3 }\ log\ (\frac{3 + u}{3 - u})] +c\]
\[=\frac{1}{18}\ [log\ (\frac{3 + (3x - 2)}{3 - (3x - 2)})] +c\]
\[=\frac{1}{18}\ [log\ (\frac{3 + 3x - 2}{3 - 3x + 2})] +c\]
\[=\frac{1}{18}\ [log\ (\frac{1 + 3x }{5 - 3x })] +c\]

Example :

\[Evaluate:\ \int \frac{dx}{{\sqrt{9 - x^2}}}\]

Soln:

\[\int \frac{dx}{{\sqrt{9 - x^2}}} = \int \frac{dx}{{\sqrt{3^2 - x^2}}} \]
\[W.K.T\ \int \frac{dx}{{\sqrt{a^2 - x^2}}} = {sin}^{-1}(\frac{x}{a})\ + c\]
\[\int \frac{dx}{{\sqrt{9 - x^2}}} = {sin}^{-1}(\frac{x}{3})\ + c\]

Example :

\[Evaluate:\ \int \frac{dx}{{\sqrt{5 - 4x^2}}}\]

Soln:

\[\int \frac{dx}{{\sqrt{5 - 4x^2}}} = \frac{1}{2}\ \int \frac{dx}{{\sqrt{\frac{5}{4} - x^2}}}\]
\[= \frac{1}{2}\ \int \frac{dx}{{\sqrt{(\frac{\sqrt{5}}{2})^2 - x^2}}}\]
\[W.K.T\ \int \frac{dx}{{\sqrt{a^2 - x^2}}} = {sin}^{-1}(\frac{x}{a})\ + c\]
\[\int \frac{dx}{{\sqrt{5 - 4x^2}}} = \frac{1}{2}\ {sin}^{-1}(\frac{x}{\frac{\sqrt{5}}{2}})\ + c\]
\[= \frac{1}{2}\ {sin}^{-1}(\frac{2x}{\sqrt{5}})\ + c\]
\[\int \frac{dx}{{\sqrt{5 - 4x^2}}} = \frac{1}{2}\ {sin}^{-1}(\frac{2x}{\sqrt{5}})\ + c\]

Example :

\[Evaluate:\ \int {\sqrt{9x^2 +16}}\ dx\]

Soln:

\[Evaluate:\ \int {\sqrt{9x^2 +16}}\ dx =3 \int {\sqrt{x^2 +\frac{16}{9}}}\ dx = 3 \int {\sqrt{x^2 + (\frac{4}{3}}})^2\ dx\]
\[\int {\sqrt{x^2 + a^2}}\ dx = \frac{x}{2}\ {\sqrt{x^2 + a^2}}\ + \frac{a^2}{2} {sin\ h}^{-1}(\frac{x}{a})\ + c\]
\[\int {\sqrt{9x^2 +16}}\ dx = 3[\frac{x}{2}\ {\sqrt{x^2 + \frac{16}{9}}}\ + \frac{16}{18} {sin\ h}^{-1}(\frac{x}{\frac{4}{3}})]\ + c\]
\[ = 3[\frac{x}{2}\ {\sqrt{x^2 + \frac{16}{9}}}\ + \frac{8}{9} {sin\ h}^{-1}(\frac{3x}{4})]\ + c\]