2.1
[tex]\int (Inx)^ndx[/tex]
This can be written as :
[tex]\int 1.(Inx)^ndx[/tex]
The formula for the integration by part is given by:
[tex]\text{Udv}=uv-\int Vdu[/tex]
Let U = (Inx)ⁿ and dv = 1
We are going to differentiate U and integrate dv ( with respect to x)
That is;
[tex]\begin{gathered} \frac{du}{dx}=(Inx)^{n-1}.\frac{1}{x} \\ \\ \Rightarrow du=n(Inx)^{n-1}\text{.}\frac{1}{x}dx \end{gathered}[/tex][tex]\begin{gathered} dv=1dx \\ \int 1dx\text{ = x} \end{gathered}[/tex]
We can now proceed to substitute into the formula;
[tex]\int (Inx)^n.1dx=^{}(Inx)^n.x-\int n(Inx)^{n-1}dx[/tex]
We can bring out n on the right-hand side since it is a constant.
[tex]\int (Inx)^ndx=x(Inx)^n-n\int (Inx)^{n-1}dx[/tex]
