Question

sum
$\int \: ( ({a}^{x}))^{2} - ({b}^{x})^{2} \div a ^{x} {b}^{x}$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Identities Used :-

$\red{ \boxed{ \sf{ {( {x}^{m}) }^{n} = {x}^{mn} \: }}}$

$\red{ \boxed{ \sf{ {x}^{m} \div {x}^{n} \: = \: {x}^{m - n} }}}$

$\red{ \boxed{ \sf{ \: \displaystyle\int\tt {a}^{x} dx = \frac{ {a}^{x} }{loga} + c}}}$

Let's solve the problem now!!!

Given integral is

$\rm :\longmapsto\:\displaystyle\int\tt \dfrac{ {( {a}^{x} )}^{2} - {( {b}^{x} )}^{2} }{ {a}^{x} {b}^{x} } \: dx$

Can be rewritten as

$\rm :\longmapsto\:\displaystyle\int\tt \dfrac{ {( {a}^{2x} )} - {( {b}^{2x} )} }{ {a}^{x} {b}^{x} } \: dx$

$\rm \: \: = \: \displaystyle\int\tt \dfrac{ {a}^{2x} }{ {a}^{x} {b}^{x} } dx - \displaystyle\int\tt \dfrac{ {b}^{2x} }{ {a}^{x} {b}^{x} } dx$

$\rm \: \: = \: \displaystyle\int\tt \dfrac{ {a}^{2x - x} }{{b}^{x} } dx - \displaystyle\int\tt \dfrac{ {b}^{2x - x} }{ {a}^{x}} dx$

$\rm \: \: = \: \displaystyle\int\tt \dfrac{ {a}^{x} }{{b}^{x} } dx - \displaystyle\int\tt \dfrac{ {b}^{x} }{ {a}^{x}} dx$

$\rm \: \: = \: \displaystyle\int\tt {\bigg(\dfrac{a}{b} \bigg) }^{x}dx - \displaystyle\int\tt {\bigg(\dfrac{b}{a} \bigg) }^{x}dx$

$\rm \: \: = \: \dfrac{{\bigg(\dfrac{a}{b} \bigg) }^{x}}{log{\bigg(\dfrac{a}{b} \bigg) }} - \dfrac{{\bigg(\dfrac{b}{a} \bigg) }^{x}}{log{\bigg(\dfrac{b}{a} \bigg) }} + c$

Additional Information :-

$\red{ \boxed{ \sf{ \displaystyle\int\tt kdx\: = x + c }}}$

$\red{ \boxed{ \sf{ \displaystyle\int\tt {e}^{x} dx\: = {e}^{x} + c }}}$

$\red{ \boxed{ \sf{ \displaystyle\int\tt \frac{1}{x}dx \: = logx + c}}}$

$\red{ \boxed{ \sf{\displaystyle\int\tt \frac{1}{ \sqrt{x} } dx\: = 2 \sqrt{x} + c }}}$

$\red{ \boxed{ \sf{\displaystyle\int\tt k \: f(x) \: dx \: = k \: \displaystyle\int\tt f(x) \: dx}}}$

Answer 2

Final answer:

$\bf \: \: \dfrac{{\bigg(\dfrac{a}{b} \bigg) }^{x}}{log{\bigg(\dfrac{a}{b} \bigg) }} - \dfrac{{\bigg(\dfrac{b}{a} \bigg) }^{x}}{log{\bigg(\dfrac{b}{a} \bigg) }} + c$

Step-by-step explanation:

$\bf \:\displaystyle\int\tt \dfrac{ {( {a}^{x} )}^{2} - {( {b}^{x} )}^{2} }{ {a}^{x} {b}^{x} } \: dx$

$\bf \: \: → \: \displaystyle\int\tt \dfrac{ {( {a}^{2x} )} - {( {b}^{2x} )} }{ {a}^{x} {b}^{x} } \: dx$

$\bf \: \: → \: \displaystyle\int\tt \dfrac{ {a}^{2x} }{ {a}^{x} {b}^{x} } dx - \displaystyle\int\tt \dfrac{ {b}^{2x} }{ {a}^{x} {b}^{x} } dx$

$\bf \: \: → \: \displaystyle\int\tt \dfrac{ {a}^{2x - x} }{{b}^{x} } dx - \displaystyle\int\tt \dfrac{ {b}^{2x - x} }{ {a}^{x}} dx$

$\bf \: \: → \: \displaystyle\int\tt \dfrac{ {a}^{x} }{{b}^{x} } dx - \displaystyle\int\tt \dfrac{ {b}^{x} }{ {a}^{x}} dx$

$\bf \: \: → \: \displaystyle\int\tt {\bigg(\dfrac{a}{b} \bigg) }^{x}dx - \displaystyle\int\tt {\bigg(\dfrac{b}{a} \bigg) }^{x}dx$

$\bf \: \: → \: \dfrac{{\bigg(\dfrac{a}{b} \bigg) }^{x}}{log{\bigg(\dfrac{a}{b} \bigg) }} - \dfrac{{\bigg(\dfrac{b}{a} \bigg) }^{x}}{log{\bigg(\dfrac{b}{a} \bigg) }} + c$