Chapter 5.3, Problem 3E

(a)

To determine

To find: The value of integral.

Answer to Problem 3E

The value is ${{\displaystyle \int }}_1^{2}f(u)du=5$

Explanation of Solution

Given information: The expression is ${{\displaystyle \int }}_1^{2}f(u)du$

Calculation:

The given expression is ${{\displaystyle \int }}_1^{2}f(u)du$

Keep in mind that changing the variable while retaining the same integration constraints will not affect the definite integral's value.

Therefore,

  $\begin{array}{c}{{\displaystyle \int }}_1^{2}f(u)du={{\displaystyle \int }}_1^{2}f(x)dx\\ =5\end{array}$

Therefore the required integral of the given expression ${{\displaystyle \int }}_1^{2}f(u)du=5$

(b)

To determine

To find: The value integral.

Answer to Problem 3E

The value is $5\sqrt{3}$

Explanation of Solution

Given information: The expression is ${{\displaystyle \int }}_1^2\sqrt{3}f(z)dz$

Calculation:

The given expression is ${{\displaystyle \int }}_1^2\sqrt{3}f(z)dz$

The integral:

  $\begin{array}{c}\sqrt{3}{{\displaystyle \int }}_1^{2}f(x)dx=\sqrt{3}\times 5\\ =5\sqrt{3}\end{array}$

Therefore the required integral of the given expression is $5\sqrt{3}$

(c)

To determine

To find: The value integral.

Answer to Problem 3E

The value is $-5$ .

Explanation of Solution

Given information: The expression is ${{\displaystyle \int }}_2^{1}f(t)dt$

Calculation:

The given expression is ${{\displaystyle \int }}_2^{1}f(t)dt$

The integral:

  $-{{\displaystyle \int }}_1^{2}f(x)dx=-5$

Therefore the required integral of the given expression is $-5$ .

(d)

To determine

To find: The value integral.

Answer to Problem 3E

The value is $-5$ .

Explanation of Solution

Given information: The expression is ${{\displaystyle \int }}_1^2[-f(x)]dx$

Calculation:

The given expression is ${{\displaystyle \int }}_1^2[-f(x)]dx$

The integral:

  $-{{\displaystyle \int }}_1^{2}f(x)dx=-5$

Therefore the required integral of the given expression is $-5$ .