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Leya [2.2K]
2 years ago
13

Find the integral ∫√(9+x)/(9-x)

Mathematics
1 answer:
densk [106]2 years ago
6 0

I suppose you mean

\displaystyle \int \frac{\sqrt{9+x}}{9-x} \, dx

Substitute y = √(9 + x). Solving for x gives x = y² - 9, so that 9 - x = 18 - y², and we have differential dx = 2y dy. Replacing everything in the integral gives

\displaystyle \int \frac{2y^2}{18 - y^2} \, dy

Simplify the integrand by dividing:

\dfrac{2y^2}{18 - y^2} = -2 + \dfrac{36}{18 - y^2}

\implies \displaystyle \int \left(\frac{36}{18-y^2} - 2\right) \, dy

For the first term of this new integral, we have the partial fraction expansion

\dfrac1{18 - y^2} = \dfrac1{\sqrt{72}} \left(\dfrac1{\sqrt{18}-y} + \dfrac1{\sqrt{18}+y}\right)

\implies \displaystyle \frac{36}{\sqrt{72}} \int \left(\frac1{\sqrt{18}-y} + \frac1{\sqrt{18}+y}\right) \, dy - 2 \int dy

The rest is trivial:

\displaystyle \sqrt{18} \int \left(\frac1{\sqrt{18}-y} + \frac1{\sqrt{18}+y}\right) \, dy - 2 \int dy

= \displaystyle \sqrt{18} \left(\ln\left|\sqrt{18}+y\right| - \ln\left|\sqrt{18}-y\right|\right) - 2y + C

= \displaystyle \sqrt{18} \ln\left|\frac{\sqrt{18}+y}{\sqrt{18}-y}\right| - 2y + C

= \boxed{\displaystyle \sqrt{18} \ln\left|\frac{\sqrt{18}+\sqrt{9+x}}{\sqrt{18}-\sqrt{9+x}}\right| - 2\sqrt{9+x} + C}

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Find the value of x 117,145,133,140,119,125
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