The question asks for the value of the integral $I = \int\sqrt{a^2 – x^2} \, dx$. This is a standard integral form in calculus.
Integral Formula Application
The standard formula for integrating a function of the form $\sqrt{a^2 - x^2}$ is:
$ \int \sqrt{a^2 - x^2} \, dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C $
Where $C$ is the constant of integration.
Result Comparison
We can rewrite the formula by factoring out $0.5$ (which is equivalent to dividing by 2):
$ 0.5 \left[ x\sqrt{a^2 – x^2} + a^2\sin^{-1}\left(\frac{x}{a}\right) \right] + C $
Comparing this result with the given options:
- Option 1: $0.5 [\sqrt{a^2 – x^2} + \sin^{-1} (\frac{x}{a})]$
- Option 2: $0.5 [x\sqrt{a^2 - x^2} + \sin^{-1} (\frac{x}{a})]$
- Option 3: $0.5 [\sqrt{a^2 - x^2} + a^2 \sin^{-1} (\frac{x}{a})]$
- Option 4: $0.5 [x\sqrt{a^2 – x^2} + a^2\sin^{-1} (\frac{x}{a})]$
The derived formula perfectly matches Option 4.
Important Questions from Indefinite Integrals
- What is \(U(x)\,V(x)\) equal to?
$X^4 + C$ is the general integral of
What is \(\rm \int \frac{dx}{x(x^2 + 1)}\) equal to?
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What is \(\smallint {{\rm{e}}^{\sin {\rm{x}}}}\left( {\frac{{{\rm{x}}{{\cos }^3}{\rm{x}} - \sin {\rm{x}}}}{{{{\cos }^2}{\rm{x}}}}} \right){\rm{dx}}\) equal to?
