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Factorization

Q. Evaluate :...

Question

Q. Evaluate : $\int_{}^{} \frac{6 x + 7}{\sqrt{x - 5) (x - 4)}} d x$

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Solution

Dear Student,

$I = \int \frac{6 x + 7}{\sqrt{(x - 5) (x - 4)}} d x = \int \frac{6 x + 7}{\sqrt{x^{2} - 9 x + 20}} d x = \int \frac{3 \times (2 x - 9) + 34}{\sqrt{x^{2} - 9 x + 20}} d x = \int \frac{3 \times (2 x - 9)}{\sqrt{x^{2} - 9 x + 20}} d x + \int \frac{34}{\sqrt{x^{2} - 9 x + 20}} d x = 3 \times \int \frac{(2 x - 9)}{\sqrt{x^{2} - 9 x + 20}} d x + 34 \times \int \frac{1}{\sqrt{x^{2} - 9 x + 20}} d x s o l v i n g 3 \times \int \frac{(2 x - 9)}{\sqrt{x^{2} - 9 x + 20}} d x l e t x^{2} - 9 x + 20 = u, t h a n d x = \frac{d u}{2 x - 9} 3 \times \int \frac{(2 x - 9)}{\sqrt{x^{2} - 9 x + 20}} d x = 3 \times \int \frac{(2 x - 9)}{\sqrt{u}} \times \frac{d u}{2 x - 9} = 3 \times \int \frac{d u}{\sqrt{u}} a p p l y p o w e r r u l e \int u^{n} d u = \frac{u^{n + 1}}{n + 1} w e g e t 3 \times \frac{u^{1 - \frac{1}{2}}}{1 - \frac{1}{2}} = 3 \times \frac{\sqrt{u}}{\frac{1}{2}} = 6 \sqrt{u} r e p l a c e u = x^{2} - 9 x + 20 = 6 \sqrt{x^{2} - 9 x + 20} n o w i n t e g r a t e 34 \times \int \frac{1}{\sqrt{x^{2} - 9 x + 20}} d x = 34 \times \int \frac{1}{\sqrt{{(x - \frac{9}{2})}^{2} - \frac{1}{4}}} d x = 34 \times \int \frac{1}{\sqrt{{(\frac{2 x - 9}{2})}^{2} - \frac{1}{4}}} d x l e t 2 x - 9 = u, t h e n d x = \frac{d u}{2} = 34 \times \int \frac{1}{\sqrt{{(\frac{u}{2})}^{2} - \frac{1}{4}}} \times \frac{d u}{2} = 34 \times \int \frac{1}{\frac{1}{2} \sqrt{u^{2} - 1}} \times \frac{d u}{2} = 34 \times \int \frac{1}{\sqrt{u^{2} - 1}} d u n o w t h i s i s a s \tan d a r d i n t e g r a l w e k n o w \int \frac{1}{\sqrt{u^{2} - 1}} d u = \log (\sqrt{u^{2} - 1} + u) h e n c e w e g e t 34 \times \log (\sqrt{u^{2} - 1} + u) r e p l a c e u = 2 x - 9, t h e a b o v e w e g e t = 34 \times \log (\sqrt{{(2 x - 9)}^{2} - 1} + 2 x - 9) a d d i n g b o t h i n t e g r a l v a l u e s w e g e t 3 \times \int \frac{(2 x - 9)}{\sqrt{x^{2} - 9 x + 20}} d x + 34 \times \int \frac{1}{\sqrt{x^{2} - 9 x + 20}} d x = 34 \times \log (\sqrt{{(2 x - 9)}^{2} - 1} + 2 x - 9) + 6 \sqrt{x^{2} - 9 x + 20} + C = 34 \times \log (\sqrt{(2 x - 9 - 1) (2 x - 9 + 1)} + 2 x - 9) + 6 \sqrt{x^{2} - 9 x + 20} + C = 34 \times \log (\sqrt{(2 x - 10) (2 x - 8)} + 2 x - 9) + 6 \sqrt{x^{2} - 9 x + 20} + C = 34 \times \log (\sqrt{4 (x - 5) (x - 4)} + 2 x - 9) + 6 \sqrt{x^{2} - 9 x + 20} + C = 34 \times \log (2 \sqrt{(x - 5) (x - 4)} + 2 x - 9) + 6 \sqrt{x^{2} - 9 x + 20} + C$
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