Q. 49 - Calculus

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.

${\int }_5^2\left(9+10x-x^2\right)dx$

Step-by-Step Solution

Answer

The exact value of definite integral is $-93$. 

Step 1. Given Information

We are given, 

${\int }_5^2\left(9+10x-x^2\right)dx$

Step 2. Finding the Integral

The definite integral is given by,

 $$\begin{gathered} {\int }_5^2\left(9+10x-x^2\right)dx={\int }_5^{2}9dx+{\int }_5^{2}10xdx-{\int }_5^2{x}^{2}dx \\ =9[2-5][phone]\frac{\mathrm{}}{\mathrm{}}\left[{\mathrm{}}^{\mathrm{}}{\mathrm{}}^{\mathrm{}}\right]\frac{\mathrm{}}{\mathrm{}}\left[{\mathrm{}}^{\mathrm{}}{\mathrm{}}^{\mathrm{}}\right] \\ =9(-3)+5(4-25)-\frac{1}{3}(8-125) \\ =-27-105+\frac{117}{3} \\ =-132+39 \\ =-93 \end{gathered}$$