Question

Find: \[ \int \frac{\sin^{-1} \left( \frac{x}{\sqrt{a + x}} \right)}{ \sqrt{a + x}} \, dx \]

Hint:

When facing integrals with inverse trigonometric functions, substituting the argument of the inverse function often helps simplify the expression. Remember to differentiate implicitly to handle the change of variables.

Answer 1

We begin by letting the given integral be: \[ I = \int \frac{\sin^{-1} \left( \frac{x}{\sqrt{a + x}} \right)}{ \sqrt{a + x}} \, dx \] To simplify this, let us use the substitution: \[ u = \sin^{-1} \left( \frac{x}{\sqrt{a + x}} \right) \] Then: \[ \sin(u) = \frac{x}{\sqrt{a + x}} \] Square both sides to eliminate the square root: \[ \sin^2(u) = \frac{x^2}{a + x} \] Now differentiate both sides with respect to \( x \): \[ 2 \sin(u) \cos(u) \frac{du}{dx} = \frac{2x}{a + x} - \frac{x^2}{(a + x)^2} \] Now, express the integrand using this substitution and simplify further to solve the integral.