Answer:
See explanation
Step-by-step explanation:
Given
[tex]\int\limits^4_2 {f(x)} \, dx =4\\ \\\int\limits^7_2 {f(x)} \, dx =-5\\ \\\int\limits^7_2 {g(x)} \, dx =2[/tex]
a. [tex]\int\limits^a_a {f(x)} \, dx =0\Rightarrow \int\limits^4_4 {f(x)} \, dx =0[/tex]
b. [tex]\int\limits^a_b {g(x)} \, dx =-\int\limits^b_a {g(x)} \, dx\Rightarrow \int\limits^2_7 {g(x)} \, dx=-\int\limits^7_2 {g(x)} \, dx=-2[/tex]
c. [tex]\int\limits^a_b {kg(x)} \, dx =k\int\limits^a_b {g(x)} \, dx \Rightarrow \int\limits^7_2 {4g(x)} \, dx =4\int\limits^7_2 {g(x)} \, dx =4\cdot 2=8[/tex]
d. [tex]\int\limits^a_b {x} \, dx +\int\limits^c_a {x} \, dx =\int\limits^c_b {x} \, dx \Rightarrow \int\limits^4_2 {f(x)} \, dx +\int\limits^7_4 {f(x)} \, dx =\int\limits^7_2 {f(x)} \, dx[/tex]
Then
[tex]\int\limits^7_4 {x} \, dx =\int\limits^7_2 {f(x)} \, dx -\int\limits^4_2 {x} \, dx =-5-4=-9[/tex]
e. [tex]\int\limits^a_b {(gf(x)-f(x))} \, dx =\int\limits^a_b {g(x)} \, dx -\int\limits^a_b {f(x)} \, dx \Rightarrow[/tex]
[tex]\int\limits^7_2 {(g(x)-f(x))} \, dx =\int\limits^7_2 {g(x)} \, dx -\int\limits^7_2 {f(x)} \, dx =2-(-5)=7[/tex]
