Q. 19 - Calculus

Question

Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals in Exercises 19–64. Use a graph to check your answer. (Hint: The integrands that involve absolute values will have to be considered piecewise.) 

${\int }_{-1}^1\left(x^4+3x+1\right)dx$

Step-by-Step Solution

Answer

The required value is $\frac{12}{5}$.

Step 1. Given Information

We are given the definite integral ${\int }_{-1}^1\left(x^4+3x+1\right)dx$ and we need to use the Fundamental Theorem of Calculus to find the exact value of the integral.      

Step 2. Finding the integral

The required value is:   

$$\begin{gathered} {\int }_{-1}^1\left(x^4+3x+1\right)dx \\ ={\int }_{-1}^1{x}^{4}dx+{\int }_{-1}^{1}3xdx+{\int }_{-1}^{1}dx \\ ={\left[\frac{x^5}{5}\right]}_{-1}^1+3{\left[\frac{x^2}{2}\right]}_{-1}^1+{\left[x\right]}_{-1}^1 \\ =\left[\frac{1^5}{5}-\frac{{\left(-1\right)}^5}{5}\right]+3\left[\frac{1^2}{2}-\frac{{\left(-1\right)}^2}{2}\right]+\left[1-\left(-1\right)\right] \\ =\frac{1}{5}+\frac{1}{5}+3\left(\frac{1}{2}-\frac{1}{2}\right)+2 \\ =\frac{2}{5}+2 \\ =\frac{12}{5} \end{gathered}$$

Step 3. Rechecking solution

The required graph is:

After plotting the graph we can see that the area under the graph is exactly same as the area obtained from the definite integral. The area under the graph is $\frac{12}{5}$ square units.