Does function $$f(x,y,z) = \frac{1}{\|\mathbf{x}\|}+\sin(x^2y)\cos(z-7)$$ belong to space $L^k(B)$ for $k = 1, 2, 3, 4$?
where $B(0,1)$ is a sphere in $\Bbb{R}^3$ centered at origin with radius $1$, and $\|\cdot\|$ is euclidean norm in $\Bbb{R}^3$.
I know that $f \in L^k(B)$ if $$\int_B |f(x,y,z)|^k \, ds < \infty$$
and since I am integrating over a sphere I use spherical coordinates to solve the integral.
$$\int_0^{2\pi}\int_0^\pi\int_0^1 \left(\frac{1}{r} + \sin(r^3\sin^3\theta\cos^2\phi\sin\phi)\cos(r\cos\theta-7) \right)^k r^2\sin\theta \, dr \, d\theta \, d\phi$$
but now I am left with some complicated integrals so my questions are, is this a correct approach? And isn't there a better one?
(I haven't found similar solved problems in textbooks nor on the internet so I would also appreciate some resources or links)