What did I do wrong while trying to solve this integral?

Your process of integrating the given function using integration by parts is absolutely correct.

Now coming to the constant of integration '$c$' - $c$ is a constant we have defined for mathematical consistency. We have to include $c$ while performing indefinite integration since there exist an infinite number of potential functions with the same derivative as the function in the integrand. So $c$ itself can take an infinite number of values.

Hence, the required factor of $25/48$ can very well be incorporated into $c$.

Hope this helped!

Whenever you are integrating a function $f(x)$ you actually have two options. You are either performing a definite integral like : $$A=\int_{a}^{b} f(x)dx$$where $A$ is actually the area under the curve of $f(x)$ with the $X$ Axis from the interval $[a,b]$ . Definite integral gives you area.

If you are performing an indefinite integral like: $$\int f(x)dx = g(x)$$ You are actually trying to find a function $g(x)$ which on differentiation will give you $f(x)$. Now the beauty of this is that not only $g(x)$ but $g(x)+1$, $g(x)+ \pi$ and all functions of the form $g(x)$ when differentiated will give the function $f(x)$. So in general the answer is

$$\int f(x) = g(x) +c$$

which is nothing but a family of curves.

Hope this helps ...