Question

Calculate the exact value of each definite integral in Exercises 47–52 by using properties of definite integrals and the formulas in Theorem 4.13.

${\int }_1^3(x+1{)}^{2}dx$

Short answer

The exact value of definite integral is $\frac{56}{3}$.

Step-by-step solution

Step 1. Given Information  

We are given,

${\int }_1^3(x+1{)}^{2}dx$

Step 2. Finding the Integral  

The definite integral is given by,

$$\begin{gathered} {\int }_1^3(x+1{)}^{2}dx={\int }_1^3{x}^2+2x+1dx \\ ={\int }_1^3{x}^{2}dx+2{\int }_1^{3}x+{\int }_1^{3}dx \\ =\frac{1}{3}\left[3^3-1\right]+2\left(\frac{1}{2}\right)\left[3^2-1\right]+[3-1] \\ =\frac{26}{3}+8+2 \\ =\frac{26+24+6}{3} \\ =\frac{56}{3} \end{gathered}$$