∫ x ^ 2 / x ^ 2 a ^ 2 dx Let x=asinθ ⇒ 1=acosθ dθ #160; / dx ⇒ dx=acosθ dθ⟶ (1) ∵ x ^ 2 / x ^ 2 a ^ 2 = a ^ 2 sin ^ 2 θ #16 ...

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Integration of Trigonometric Functions

Solve: ∫x2x2...

Question

Solve:
$\int \frac{x^{2}}{x^{2} - a^{2}} d x$

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Solution

$\int \frac{x^{2}}{x^{2} - a^{2}} d x L e t x = a s i n θ \Rightarrow 1 = a c o s θ \frac{d θ}{d x} \Rightarrow d x = a c o s θ d θ ⟶ (1) ∵ \frac{x^{2}}{x^{2} - a^{2}} = \frac{a^{2} {s i n}^{2} θ}{a^{2} {s i n}^{2} θ - a^{2}} = - {t a n}^{2} θ (∵ 1 - {s i n}^{2} θ = {c o s}^{2} θ a n d \frac{s i n θ}{c o s θ} = t a n θ) ∴ \int \frac{x^{2}}{x^{2} - a^{2}} d x = \int (- {t a n}^{2} θ) . a c o s θ d θ = - a \int ({s e c}^{2} θ - 1) c o s θ d θ (∵ 1 + {t a n}^{2} θ = {s e c}^{2} θ) = a \int (c o s θ - s e c θ) d θ = a s i n θ - l o g | s e c θ + t a n θ | + C (∵ \int s e c θ = l o g | s e c θ + t a n θ |) ∵ x = a s i n θ \Rightarrow s i n θ = \frac{x}{a} ∴ \int \frac{x^{2}}{x^{2} - a^{2}} d x = a . \frac{x}{a} - l o g ∣ ∣ ∣ \frac{a}{\sqrt{a^{2} - x^{2}}} + \frac{x}{\sqrt{a^{2} - x^{2}}} ∣ ∣ ∣ + C (f r o m t h e f i g u r e g e t t h e v a l u e o f s e c θ a n d t a n θ) = x - l o g ∣ ∣ ∣ \frac{a + x}{\sqrt{a^{2} - x^{2}}} ∣ ∣ ∣ + C$

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