If ∫$$dt3at$$−$$2t2=axsin$$−$$1$$⁡$$t2a2$$−$$1$$ the value of x is $$_________$$

$$t2a2$$−$$1$$ is dimensionless ±$$t=$$±a[a]$$=$$[t]$$3at$$−$$2t2=$$[t]$$dt3at$$−$$2t2=$$[t][t]$$=M0L0T0ax$$ should be dimensionless, so $$x=0$$ .

Dimensions of physical quantities

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Question

If $\int \frac{d t}{\sqrt{3 a t - 2 t^{2}}} = a^{x} \sin^{- 1} (\frac{t^{2}}{a^{2}} - 1)$ the value of x is _________

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Solution

$(\frac{t^{2}}{a^{2}} - 1)$ is dimensionless $\pm t = \pm a$
$\begin{array}{l} [a] = [t] \\ \sqrt{(3 a t - 2 t^{2})} = [t] \\ [\frac{d t}{\sqrt{3 a t - 2 t^{2}}}] = \frac{[t]}{[t]} = [M^{0} L^{0} T^{0}] \\ a^{x} should be dimensionless, so x = 0 . \end{array}$

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