Chapter 6.4, Problem 67E

To determine

To calculate:The value of f(5)

Answer to Problem 67E

The value of f (5) is option (D) 4.555

Explanation of Solution

Given information:

Given the function is $f(x)={\displaystyle \underset{a}{\overset{x}{\int }}\ln (2+\sin t)dt}$

Formula used:

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx+{\displaystyle \underset{b}{\overset{c}{\int }}f\left(x\right)}dx={\displaystyle \underset{a}{\overset{c}{\int }}f\left(x\right)}dx$

Calculation:

Correct option is (D).

Since,

  $f\left(5\right)={\displaystyle \underset{a}{\overset{5}{\int }}\ln \left(2+\sin t\right)}dt$

Since we know that

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx+{\displaystyle \underset{b}{\overset{c}{\int }}f\left(x\right)}dx={\displaystyle \underset{a}{\overset{c}{\int }}f\left(x\right)}dx$

So,

  $\begin{array}{l}f\left(5\right)={\displaystyle \underset{a}{\overset{3}{\int }}\ln \left(2+\sin t\right)dt+}{\displaystyle \underset{3}{\overset{5}{\int }}\ln \left(2+\sin t\right)dt}\\ =f\left(3\right)+\text{NINT}\left(\ln \left(2+\sin t\right),t,3,5\right)\\ =4+0.555\\ f\left(5\right)=4.555\end{array}$

Conclusion:

The value of f (5) =4.555