Question

$\int 12x^{ \frac{7}{2} }~dx=...$

Answer 1

Factor out the constant and use power rule : $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$

$\int 12x^{7/2}\,dx =12\int x^{7/2}\,dx = 12\left(\frac{x^{7/2+1}}{7/2+1}\right)+C=\frac{8x^{9/2}}{3}+C$

Answer 2

This question is type of
integration X^n dx= x^n+1/n+1 +c    where c is arbitrary const.
integration 12 x^7/2 dx
12 integration x^7/2 +1                                            no effect on constant during
                       7/2 +1
integration     

12 x^9/2    ⇒8/3 x^9/2 +c
9/2