# Integration of polynomial 2.

#### paulmdrdo

##### Active member
3.) ∫(3+s)1/2(s+1)2ds

#### MarkFL

Staff member
I would let:

$$\displaystyle u=s+3\,\therefore\,du=ds$$

and now we have:

$$\displaystyle \int u^{\frac{1}{2}}(u-2)^2\,du$$

Now, expand, distribute, and then apply the power rule term by term.

#### paulmdrdo

##### Active member
I would let:

$$\displaystyle u=s+3\,\therefore\,du=ds$$

and now we have:

$$\displaystyle \int u^{\frac{1}{2}}(u-2)^2\,du$$

Now, expand, distribute, and then apply the power rule term by term.
how do you get (u-2)2?

#### Sudharaka

##### Well-known member
MHB Math Helper
Hi everyone, An alternative method without using substitutions is to write the integrand only using $$s+3$$.

\begin{eqnarray}

\int(s+3)^{1/2}(s+1)^2\,ds&=&\int(s+3)^{1/2}(s+3-2)^2\,ds\\

&=&\int(s+3)^{1/2}\left((s+3)^2-4(s+3)+4\right)\,ds\\

&=&\int(s+3)^{5/2}\,d(s+3)-4\int(s+3)^{3/2}\,d(s+3)+4\int(s+3)^{1/2}\,d(s+3)\\

\end{eqnarray}

Hope you can continue. #### Sudharaka

##### Well-known member
MHB Math Helper
how do you get (u-2)2?
Hi paulmdrdo, The $$s+1$$ in the integrand becomes $$u-2$$. That is, $$u=s+3\Rightarrow u-2=s+1$$.