Tan -13 a2 x-x3-a2-3 x2- a-0 - -2-3- x -a-3

Let x=a tanθ ⇒ x/a=tanθ⇒θ tan^ 1 ( x/a ) ∴ tan^ 1 ( 3a^2x x^3/a^3 3ax^2 )=tan^ 1 ( 3a^2(a tan θ) (a tan θ)^3/a^3 3a(a tan θ)^2 ) =tan^ 1 ( a^3(3tan θ) tan^3θ ...

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Byju's Answer

Standard XII

Mathematics

Integration Using Substitution

tan -13 a2 x-...

Question

$t a n^{- 1} (\frac{3 a^{2} x - x^{3}}{a^{3} - 3 a x^{2}}), a > 0; \frac{- a}{\sqrt{3}} \leq x \leq \frac{a}{\sqrt{3}}$

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Solution

Let $x = a t a n θ$
$\Rightarrow \frac{x}{a} = t a n θ \Rightarrow θ t a n^{- 1} (\frac{x}{a}) ∴ t a n^{- 1} (\frac{3 a^{2} x - x^{3}}{a^{3} - 3 a x^{2}}) = t a n^{- 1} (\frac{3 a^{2} (a t a n θ) - (a t a n θ)^{3}}{a^{3} - 3 a (a t a n θ)^{2}}) = t a n^{- 1} (\frac{a^{3} (3 t a n θ) - t a n^{3} θ}{a^{3} (1 - 3 t a n^{2} θ)}) = t a n^{- 1} (\frac{3 t a n θ - t a n^{3} θ}{1 - 3 t a n^{2} θ}) = t a n^{- 1} (t a n 3 θ) (∵ t a n 3 θ = \frac{3 t a n θ - t a n^{3} θ}{1 - 3 t a n^{2} θ}) = 3 θ = 3 t a n^{- 1} (\frac{x}{a})$

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Similar questions

Q. Prove

{tan}^{- 1} (\frac{3 a^{2} x - x^{3}}{a^{3} - 3 a x^{3}}), a > 0, \frac{- a}{\sqrt{2}} \leq x \leq \frac{a}{\sqrt{3}}

Q. simplify

{tan}^{- 1} \frac{3 a^{2} x - x^{3}}{a (a^{2} - 3 x^{2})}

Q. Differentiate

{tan}^{- 1} (\frac{3 a^{2} x - x^{3}}{a^{3} - 3 a x^{2}})

- \frac{1}{\sqrt{3}} < \frac{x}{a} < \frac{1}{\sqrt{3}}

Q. Write the function in the simplest form:

{tan}^{- 1} (\frac{3 a^{2} x - x^{3}}{a^{3} - 3 {a x}^{2}}), a > 0; \frac{- a}{\sqrt{3}} \leq x \leq \frac{a}{\sqrt{3}}

Write the given trigonometric expression in its simplest form.

{tan}^{- 1} (\frac{3 a^{2} x - x^{3}}{a^{3} - 3 a x^{2}}), a > 0; \frac{- a}{\sqrt{3}} \leq x \leq \frac{a}{\sqrt{3}}

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tan−1(3a2x−x3a3−3ax2),a>0;−a√3≤x≤a√3

Let x=a tanθ ⇒ xa=tanθ⇒θtan−1(xa)∴ tan−1(3a2x−x3a3−3ax2)=tan−1(3a2(a tan θ)−(a tan θ)3a3−3a(a tan θ)2)=tan−1(a3(3tan θ)−tan3θa3(1−3tan2θ))=tan−1(3 tan θ−tan3θ1−3tan2θ)=tan−1(tan 3θ)(∵ tan 3θ=3tan θ−tan3θ1−3tan2θ)=3θ=3tan−1(xa)

$t a n^{- 1} (\frac{3 a^{2} x - x^{3}}{a^{3} - 3 a x^{2}}), a > 0; \frac{- a}{\sqrt{3}} \leq x \leq \frac{a}{\sqrt{3}}$