Question

Could you please help me with this problem of integration ​

Answer 1

$\displaystyle\int4(z^2+2z+1)\ dz$

Since 4 is just a constant, we can take it outside the integral.

$\displaystyle 4\int(z^2+2z+1)\ dz$

Now we can split each term as,

$\displaystyle 4\int z^2\cdot dz+4\int 2z\cdot dz+4\int 1\cdot dz$

We know  $\displaystyle\int z^n\cdot dz=\dfrac{z^{n+1}}{n+1}.$

So,

$\displaystyle 4\int z^2\cdot dz+4\int 2z\cdot dz+4\int 1\cdot dz\\ \\ \\ \Longrightarrow\ 4\cdot\dfrac{z^3}{3}+8\int z\cdot dz+4\int dz\\ \\ \\ \Longrightarrow\ \dfrac{4z^3}{3}+8\cdot\dfrac{z^2}{2}+4z\\ \\ \\ \Longrightarrow\ \dfrac{4}{3}z^3+4z^2+4z\\ \\ \\ \Longrightarrow\ 4\left(\dfrac{1}{3}z^3+z^2+z\right)$

Or we can do the following:

$\begin{aligned}&\int4(z^2+2z+1)\ dz\\ \\ \Longrightarrow\ \ &4\int(z^2+2z+1)\ dz\\ \\ \Longrightarrow\ \ &4\left(\int z^2\dcot dz+\int 2z\cdot dz+\int 1\cdot dz\right)\\ \\ \Longrightarrow\ \ &4\left(\dfrac{z^3}{3}+2\cdot\dfrac{z^2}{2}+z\right)\\ \\ \Longrightarrow\ \ &4\left(\dfrac{1}{3}z^3+z^2+z\right)\end{aligned}$

Hence integrated!

Answer 2

$\huge\underline\textbf{Answer:-}$

Refer the given attachment:-