c) \( \int \arcsin x d x \)

To solve the integral \( \int \arcsin x \, dx \), we can use integration by parts. Set \( u = \arcsin x \) (so that \( du = \frac{1}{\sqrt{1 - x^2}} dx \)) and \( dv = dx \) (hence \( v = x \)). Applying the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we get: \[ \int \arcsin x \, dx = x \arcsin x - \int x \cdot \frac{1}{\sqrt{1 - x^2}} \, dx. \] The remaining integral \( \int \frac{x}{\sqrt{1 - x^2}} \, dx \) can be solved by the substitution \( w = 1 - x^2 \), leading to the result: \[ \int \arcsin x \, dx = x \arcsin x + \sqrt{1 - x^2} + C, \] where \( C \) is the constant of integration. Voilà, you've got the solution wrapped up nicely!