Key to "Solved by the radical integration. ∫1−x2x4dx"

High School
Calculus and Analysis
Calculus 1
Integrals

Cem Hayes

Answered question

2021-05-12

Solved by the radical integration.

\int \frac{\sqrt{1 - x^{2}}}{x^{4}} d x

Answer & Explanation

Arham Warner

Skilled2021-05-13Added 102 answers

Solution:
$\int \frac{\sqrt{1 - x^{2}}}{x^{4}} d x$
Let $x = \sin θ \Rightarrow d x = \cos θ \cdot d θ$
$= \int \frac{\sqrt{1 - \sin^{2} θ}}{\sin^{4} θ} \cos θ d θ$
$= \int \frac{\cos^{2} θ}{\sin^{4} θ} d θ = \int \cot^{2} θ \cdot \csc^{2} θ d θ$
Let $u = \cot θ \Rightarrow d u = - \csc^{2} θ d θ$
$= - \int u^{2} d u$
$= - \frac{u^{3}}{3} + c$
$= - \frac{\cot^{3} θ}{3} + c$
$= - \frac{(1 - x^{2}) \sqrt{1 - x^{2}}}{3 \times 3} + c$
$\int \frac{\sqrt{1 - x^{2}}}{x^{4}} d x = - \frac{(1 - x^{2}) \sqrt{1 - x^{2}}}{3 \times 3} + c$

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