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Question: Evaluate \(\int_{}^{}\frac{xdx}{\sqrt{(7x - 10 - x^{2})^{3}}}\) where \(t = \frac{\sqrt{7x - 10 - x^...

Evaluate ∫xdx(7x−10−x2)3\int_{}^{}\frac{xdx}{\sqrt{(7x - 10 - x^{2})^{3}}} where t=7x−10−x2x−2t = \frac{\sqrt{7x - 10 - x^{2}}}{x - 2}

A

−29(−5t+2t)+c\frac{- 2}{9}\left( \frac{- 5}{t} + 2t \right) + c

B

29(−5t−2t)+c\frac{2}{9}\left( \frac{- 5}{t} - 2t \right) + c

C

−19(5t+2t)+c\frac{- 1}{9}\left( \frac{5}{t} + 2t \right) + c

D

None of these

Answer

−29(−5t+2t)+c\frac{- 2}{9}\left( \frac{- 5}{t} + 2t \right) + c

Explanation

Solution

In this case a<0a < 0 and c<0.c < 0. Therefore neither (I) nor (II) Euler substitution is applicable. But the quadratic 7x−10−x27x - 10 - x^{2} has real roots α=2,β=5.\alpha = 2,\beta = 5.

∴\mathbf{\therefore}We use the (III) i.e.,

7x−10−x2=(x−2)(5−x)=(x−2)t\sqrt{7x - 10 - x^{2}} = \sqrt{(x - 2)(5 - x)} = (x - 2)twhere (5−x)=(x−2)t2(5 - x) = (x - 2)t^{2}or 5+2t2=x(1+t2)5 + 2t^{2} = x(1 + t^{2})

∴x=5+2t21+t2\therefore x = \frac{5 + 2t^{2}}{1 + t^{2}}

⇒(x−2)t=(5+2t21+t2−2)t=3t1+t2\mathbf{\Rightarrow (x - 2)t =}\left( \frac{\mathbf{5 + 2}\mathbf{t}^{\mathbf{2}}}{\mathbf{1 +}\mathbf{t}^{\mathbf{2}}}\mathbf{- 2} \right)\mathbf{t =}\frac{\mathbf{3t}}{\mathbf{1 +}\mathbf{t}^{\mathbf{2}}}, ∴dx=−6t(1+t2)2dt\mathbf{\therefore}\mathbf{dx =}\frac{\mathbf{-}\mathbf{6t}}{\mathbf{(1 +}\mathbf{t}^{\mathbf{2}}\mathbf{)}^{\mathbf{2}}}\mathbf{dt}

Hence, I=∫xdx(7x−10−x2)3=∫(5+2t21+t2).−6t(1+t2)2dt(3t1+t2)3I = \int_{}^{}\frac{xdx}{(\sqrt{7x - 10 - x^{2}})^{3}} = \int_{}^{}\frac{\left( \frac{5 + 2t^{2}}{1 + t^{2}} \right).\frac{- 6t}{(1 + t^{2})^{2}}dt}{\left( \frac{3t}{1 + t^{2}} \right)^{3}}

=−627∫5+2t2t2dt= \frac{- 6}{27}\int_{}^{}\frac{5 + 2t^{2}}{t^{2}}dt =−29(5t2+2)dt=−29[−5t+2t]+c= \frac{- 2}{9}\left( \frac{5}{t^{2}} + 2 \right)dt = \frac{- 2}{9}\left\lbrack \frac{- 5}{t} + 2t \right\rbrack + c

∴∫xdx(7x−10−x2)3=−29(−5t+2t)+c\therefore\int_{}^{}{\frac{xdx}{(\sqrt{7x - 10 - x^{2})^{3}}} =}\frac{- 2}{9}\left( \frac{- 5}{t} + 2t \right) + c, where t=7x−10−x2x−2t = \frac{\sqrt{7x - 10 - x^{2}}}{x - 2}