Question

8) Calcule a integral

$\int\ {} \frac{dx}{x^2 \sqrt{5-x^2} } \,$

Answer 1

Oi Lucas

∫ dx/(x²*√(5 - x²))

seja x = √5*sin(u) e dx = √5*cos(u) du

√(5 - x²) = √(5 - 5sin²/u)) = √5*cos(u) e u = sin^(-1)(x/sqrt(5))

∫ dx/(x²*√(5 - x²)) : = √5 ∫ (csc²(u))/(5√5)) du =

1/5  ∫ csc²(u) du

∫ csc^2(u) é -cotg(u)

1/5  ∫ csc²(u) du   = -(cotg(u))/5 + C 

∫ dx/(x²*√(5 - x²)) = -√(5 - x²)/(5x) + C

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Answer 2

$\sf \displaystyle \int \frac{dx}{x^2\sqrt{5-x^2}}\\\\\\=\int \frac{\cot \left(u\right)}{5\sin \left(u\right)\cos \left(u\right)}du\\\\\\=\frac{1}{5}\cdot \int \frac{\cot \left(u\right)}{\sin \left(u\right)\cos \left(u\right)}du\\\\\\=\frac{1}{5}\cdot \int \csc ^2\left(u\right)du\\\\\\=\frac{1}{5}\left(-\cot \left(u\right)\right)\\\\\\=\frac{1}{5}\left(-\cot \left(\arcsin \left(\frac{1}{\sqrt{5}}x\right)\right)\right)\\\\\\=-\frac{\sqrt{5-x^2}}{5x}$

$\to \boxed{\sf =-\frac{\sqrt{5-x^2}}{5x}+C}$