Question

solve this question ... no wrong answers ​

Answer 1

QuestioN :-

$\displaystyle \int \: \sf \frac{6x + 7}{ \sqrt{(x - 5)(x - 4)} } \: dx \: = ?$

AnsweR :-

$\displaystyle \sf \int \: \frac{6x + 7}{ \sqrt{ (x - 5)(x - 4) } } \: dx \sf \: = 6 \sqrt{ (x - 5)(x - 4)} + 34 \: log \bigg \{ \bigg| x - \frac{9}{2} + \sqrt{ (x - 5)(x - 4) } \: \bigg| \bigg \} + c$

SolutioN :-

$\displaystyle \int \: \sf \frac{6x + 7}{ \sqrt{(x - 5)(x - 4)} } \: dx \: \\ \\ \int \: \sf \frac{6x + 7}{ \sqrt{ {x}^{2} - 9x + 20 } } \: dx \: \\ \\ \sf \: let \: 6x + 7 = a \bigg( \frac{d( {x}^{2} - 9x + 20) }{dx} \bigg) + b \\ \\ \sf \: 6x + 7 = a(2x - 9) + b \\ \\ \sf \:6x + 7 = 2ax - 9a + b \\ \\ \sf \: comparing \: the \: coefficients \mapsto \\ \\ \sf \star 2a = 6 \\ \\ \sf \: a = \frac{6}{2} \\ \\ \sf \: a = \frac{ { \cancel{6}} \: \: ^{3} }{ \cancel{2}} \\ \\ \boxed{ \huge \sf \: a = 3} \\ \\ \sf \star \: - 9a + b = 7 \\ \\ \sf \: - 9 \times 3 + b = 7 \\ \\ \sf \: - 27 + b = 7 \\ \\ \sf \: b = 27 + 7 \\ \\ \boxed{ \huge \: \sf b = 34}$

Now ,

$\displaystyle \int \: \sf \frac{6x + 7}{ \sqrt{( {x}^{2} - 9x + 20)} } \: dx \: = \displaystyle \int \: \sf \frac{ax + b}{ \sqrt{ {x}^{2} - 9x + 20 } } \: dx \: \\ \\ = \sf \: \displaystyle \int \: \sf \frac{3(2x - 9) + 34}{ \sqrt{ {x}^{2} - 9x + 20} } \: dx \: \\ \\ = \sf3 \int \: \frac{2x - 9}{ \sqrt{ {x}^{2} - 9x + 20 } } \: dx + \int \: \frac{34}{ \sqrt{ {x}^{2} - 9x + 20 } } \: dx \\ \\ \sf \: now \: let \: {x}^{2} - 9x + 20 = t \\ \\ \sf \: (2x - 9)dx = dt \\ \\ \sf \: so \\ \\ \sf3 \int \: \frac{2x - 9}{ \sqrt{ {x}^{2} - 9x + 20 } } \: dx = 3 \int \: \frac{dt}{ \sqrt{t} } \\ \\ \sf \: = 6 \sqrt{t} + c \\ \\ \sf3 \int \: \frac{2x - 9}{ \sqrt{ {x}^{2} - 9x + 20 } } \: dx= 6 \sqrt{ {x}^{2} - 9x + 20} + c$

$\sf \: and \\ \\ \sf \: \int \frac{34}{ \sqrt{ {x}^{2} - 9x + 20 } } \: dx \mapsto$

Now convert Denominator into complete square method →

$\sf {x}^{2} - 9x + 20 \\ \\ \sf \: = {x}^{2} - 2x \bigg( \frac{9}{2} \bigg) + \frac{81}{4} + 20 - \frac{81}{4} \\ \\ \sf \: = { \bigg(x - \frac{9}{2} \bigg)}^{2} - \bigg( { \frac{1}{2} \bigg)}^{2}$

So,

$\displaystyle \: \sf\int \: \frac{34}{ \sqrt{ {x}^{2} - 9x + 20 } } \: dx = \int \: \frac{34}{{ \big(x - \frac{9}{2} \big)}^{2} - \big( { \frac{1}{2} \big)}^{2} } \: dx$

Now, Using the formula →

$\displaystyle \sf \: \int \: \frac{1}{ \sqrt{ {x}^{2} - {a}^{2} } } \: dx= log( |x + \sqrt{ {x}^{2} - {a}^{2} } | ) + c$

$\displaystyle \sf \: \int \: \frac{34}{{ \big(x - \frac{9}{2} \big)}^{2} - \big( { \frac{1}{2} \big)}^{2} } \: dx \: = 34 \: log \bigg \{ \bigg| \big(x - \frac{9}{2} \big) + \sqrt{ { \bigg (x - \frac{9}{2} \bigg)}^{2} - { \bigg( \frac{1}{2} \bigg) }^{2} } \bigg| \bigg \} + c \\ \\ = \sf34 \: log \bigg \{ \bigg| x - \frac{9}{2} + \sqrt{ {x}^{2} - 9x + 20 } \: \bigg| \bigg \} + c$

So Far ,

$\displaystyle \sf \int \: \frac{6x + 7}{ \sqrt{ (x - 5)(x - 4) } } \: dx = 6 \sqrt{ {x}^{2} - 9x + 20} + 34 \: log \bigg \{ \bigg| x - \frac{9}{2} + \sqrt{ {x}^{2} - 9x + 20 } \: \bigg| \bigg \} + c \\ \\ \sf \: = 6 \sqrt{ (x - 5)(x - 4)} + 34 \: log \bigg \{ \bigg| x - \frac{9}{2} + \sqrt{ (x - 5)(x - 4) } \: \bigg| \bigg \} + c$