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(C) We need to evaluate the integral $I = \int \sqrt{x^2 - 8x + 7} \, dx$. First,complete the square for the quadratic expression: $x^2 - 8x + 7 = (x^

Vedclass · Accepted Answer

$\frac{1}{2}(x - 4)\sqrt{x^2 - 8x + 7} - \frac{9}{2}\log |x - 4 + \sqrt{x^2 - 8x + 7}| + c$

Vedclass · Answer

(C) We need to evaluate the integral $I = \int \sqrt{x^2 - 8x + 7} \, dx$. First,complete the square for the quadratic expression: $x^2 - 8x + 7 = (x^2 - 8x + 16) - 16 + 7 = (x - 4)^2 - 9 = (x - 4)^2 - 3^2$. Now the integral becomes $I = \int \sqrt{(x - 4)^2 - 3^2} \, dx$. Using the standard formula $\int \sqrt{t^2 - a^2} \, dt = \frac{t}{2}\sqrt{t^2 - a^2} - \frac{a^2}{2}\log |t + \sqrt{t^2 - a^2}| + c$,where $t = x - 4$ and $a = 3$: $I = \frac{x - 4}{2}\sqrt{(x - 4)^2 - 3^2} - \frac{3^2}{2}\log |(x - 4) + \sqrt{(x - 4)^2 - 3^2}| + c$. Simplifying this,we get $I = \frac{1}{2}(x - 4)\sqrt{x^2 - 8x + 7} - \frac{9}{2}\log |x - 4 + \sqrt{x^2 - 8x + 7}| + c$. Thus,the correct option is $C$.

$\int \sqrt{x^2 - 8x + 7} \, dx = $

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