Choose the correct answer in the given question-x2-8 x-7 d x is equal to - a frac12 x -4 sqrt x - 2-8 x -7-9 log- x -4- sqrt x - 2-8 x -7- Cb frac 12x-4 isqrt x- 2-8 x-7-9 log- x-4- isqrt x- 2-8 x-7-C 1c - frac 12x-4 isqrt x- 2-8 x-7-3 isqrt 2log- x-4- sqrt x- 2-8 x-7-Cd frac 12x-4 isqrt x- 2-8 x-7-frac 92log- x-4- lsqrt -x- 2-8 x-7-C-

Let I =∫√(x^2 8x +7)dx ⇒ I =∫√(x^2 8x+7+(4)^2 (4)^2)dx =∫√((x 4)^2+7 16)dx =∫√((x 4)^2 (3)^2)dx [ ∵∫√(x^2 a^2)dx =x/2√(x^2 a^2) a^2/2log |x+√(x^2 a^2)| ] ...

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Byju's Answer

Standard XII

Mathematics

Integration by Partial Fractions

Choose the co...

Question

Choose the correct answer in the given question.
$\int \sqrt{x^{2} - 8 x + 7} d x$ is equal to
$(a) \frac{1}{2} (x - 4) \sqrt{x^{2} - 8 x + 7} + 9 l o g | x - 4 + \sqrt{x^{2} - 8 x + 7} | + C (b) \frac{1}{2} (x + 4) \sqrt{x^{2} - 8 x + 7} + 9 l o g | x + 4 + \sqrt{x^{2} - 8 x + 7} | + C (c) \frac{1}{2} (x - 4) \sqrt{x^{2} - 8 x + 7} - 3 \sqrt{2} l o g | x - 4 + \sqrt{x^{2} - 8 x + 7} | + C (d) \frac{1}{2} (x - 4) \sqrt{x^{2} - 8 x + 7} - \frac{9}{2} l o g | x - 4 + \sqrt{x^{2} - 8 x + 7} | + C$

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Solution

Let $I = \int \sqrt{x^{2} - 8 x + 7} d x$
$\Rightarrow I = \int \sqrt{x^{2} - 8 x + 7 + (4)^{2} - (4)^{2}} d x = \int \sqrt{(x - 4)^{2} + 7 - 16} d x = \int \sqrt{(x - 4)^{2} - (3)^{2}} d x [∵ \int \sqrt{x^{2} - a^{2}} d x = \frac{x}{2} \sqrt{x^{2} - a^{2}} - \frac{a^{2}}{2} l o g | x + \sqrt{x^{2} - a^{2}} |] \Rightarrow I = \frac{x - 4}{2} \sqrt{x^{2} - 8 x + 7} - \frac{9}{2} l o g | x - 4 + \sqrt{x^{2} - 8 x + 7} | + C$
So, option (d)is the correct answer.

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Choose the correct answer in the given question. ∫√x2−8x+7dx is equal to (a)12(x−4)√x2−8x+7+9log|x−4+√x2−8x+7|+C(b)12(x+4)√x2−8x+7+9log|x+4+√x2−8x+7|+C(c)12(x−4)√x2−8x+7−3√2log|x−4+√x2−8x+7|+C(d)12(x−4)√x2−8x+7−92log|x−4+√x2−8x+7|+C

Let I=∫√x2−8x+7dx ⇒I=∫√x2−8x+7+(4)2−(4)2dx=∫√(x−4)2+7−16dx=∫√(x−4)2−(3)2dx[∵∫√x2−a2dx=x2√x2−a2−a22log|x+√x2−a2|]⇒I=x−42√x2−8x+7−92log|x−4+√x2−8x+7|+C So, option (d)is the correct answer.