Integrate with respect to x : (a)1/(9-8x-x^2) (b)1/(3x^2-9)

1.	Integrate with respect to x : (a)1/(9-8x-x^2) (b)1/(3x^2-9)
Answer» Answer (a) 1/10{log\|(X + 9)/(1 - x)\|} + C (B) 1/6√3{log\|(x - √3)/(x + √3)\| + C Solution $\sf (a) \: \dfrac{1}{9 - 8x - {x}^{2} }$ $\sf \underline{Integrating \: with \: respect \: to \: \bf{x} }$ $\sf \longrightarrow \int\dfrac{dx}{9 - 8x - {x}^{2} } \\ \\ \sf \longrightarrow \int \dfrac{dx}{25 - 16 - 8x - {x}^{2} } \\ \\ \sf \longrightarrow \int \dfrac{dx}{25 - (x {}^{2} + 4 {}^{2} + 2 \times 4 \times x )} \\ \\ \sf \longrightarrow \int \dfrac{dx}{5 {}^{2} - (x + 4) {}^{2} }$ Now let US CONSIDER $\sf x + 4 = t \\ \\ \sf Differentiating \: with \: respect \: to \: x \\ \\ \implies \sf \dfrac{d}{dx} (x + 4) = \frac{dt}{dx} \\ \\ \sf\implies 1 = \frac{dt}{dx} \\ \\ \sf\implies dx = dt$ THUS , the required integration becomes : $\sf \longrightarrow \int \dfrac{dt}{5^{2} - t^{2}} \\\\ \sf\longrightarrow \dfrac{1}{2\times 5 } \log \| \dfrac{t + 5}{t - 5 } \\\\ \sf\longrightarrow \dfrac{1}{10} \log\|\dfrac{ 5+x + 4 }{5 - ( x + 4) } \| + C \\\\ \sf\longrightarrow \dfrac{1}{10} \log \| \dfrac{x+9}{1-x}\| + C$ ___________________ $\sf (b) \dfrac{1}{3x^{2} - 9}$ $\sf \underline{Integrating \: we \: have : }$ $\sf\longrightarrow \int \dfrac{1}{3x^{2} - 9 } \\\\ \sf\longrightarrow \int \dfrac{1}{3(x^{2} - 3)} \\\\$ $\sf \longrightarrow \dfrac{1}{3} \int \dfrac{dx}{x^{2} - (\sqrt{3})^{2}} \\\\$ $\sf\longrightarrow \dfrac{1}{6\sqrt{3}} \log \|\dfrac{x-\sqrt{3}}{x+\sqrt{3}} \| + C$ $\bf \large\underline{Formula \: used }$ ✪ $\sf \int \dfrac{dx}{x^{2} - a^{2} } = \dfrac{1}{2a}\log \|\dfrac{x-a}{x + a} \| + C$ ✪ $\sf \int \dfrac{dx}{a^{2} - x^{2} } = \dfrac{1}{2a} \log\|\dfrac{a + x}{a - x} \| + C$

Answer»

Answer

(a) 1/10{log|(X + 9)/(1 - x)|} + C

(B) 1/6√3{log|(x - √3)/(x + √3)| + C

Solution

$\sf (a) \: \dfrac{1}{9 - 8x - {x}^{2} }$

$\sf \underline{Integrating \: with \: respect \: to \: \bf{x} }$

$\sf \longrightarrow \int\dfrac{dx}{9 - 8x - {x}^{2} } \\ \\ \sf \longrightarrow \int \dfrac{dx}{25 - 16 - 8x - {x}^{2} } \\ \\ \sf \longrightarrow \int \dfrac{dx}{25 - (x {}^{2} + 4 {}^{2} + 2 \times 4 \times x )} \\ \\ \sf \longrightarrow \int \dfrac{dx}{5 {}^{2} - (x + 4) {}^{2} }$

Now let US CONSIDER

$\sf x + 4 = t \\ \\ \sf Differentiating \: with \: respect \: to \: x \\ \\ \implies \sf \dfrac{d}{dx} (x + 4) = \frac{dt}{dx} \\ \\ \sf\implies 1 = \frac{dt}{dx} \\ \\ \sf\implies dx = dt$

THUS , the required integration becomes :

$\sf \longrightarrow \int \dfrac{dt}{5^{2} - t^{2}} \\\\ \sf\longrightarrow \dfrac{1}{2\times 5 } \log | \dfrac{t + 5}{t - 5 } \\\\ \sf\longrightarrow \dfrac{1}{10} \log|\dfrac{ 5+x + 4 }{5 - ( x + 4) } | + C \\\\ \sf\longrightarrow \dfrac{1}{10} \log | \dfrac{x+9}{1-x}| + C$