Q. 55 - 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.)

${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x\left(cs{c}^2\right(x^2\left)\right)dx$

Step-by-Step Solution

Answer

${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x\left(cs{c}^2\right(x^2\left)\right)dx=\frac{1}{2}cot\left(\frac{{\mathrm{\pi }}^2}{16}\right)-\frac{1}{2}cot\left(\frac{{\mathrm{\pi }}^2}{4}\right)$.

Step 1. Given information.

A definite integral is given as ${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x\left(cs{c}^2\right(x^2\left)\right)dx$.

Step 2. Using the Fundamental theorem of Calculus.

Let

$$\begin{gathered} f\left(x\right)=-cotx \\ g\left(x\right)=x^2 \end{gathered}$$

such that

$$\begin{gathered} f\text{'}\left(x\right)=cs{c}^{2}x \\ g\text{'}\left(x\right)=2x \\ f\text{'}\left(g\right(x\left)\right)=cs{c}^2\left(x^2\right) \end{gathered}$$

Now we get

$$\begin{gathered} {\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x\left(cs{c}^2\right(x^2\left)\right)dx \\ =\frac{1}{2}{\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}2x\left(cs{c}^2\right(x^2\left)\right)dx \\ =-\frac{1}{2}{\left[\mathrm{co}t\right(x^2\left)\right]}_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} [f\text{'}(g\left(x\right))g\text{'}(x)dx=f(g\left(x\right)\left)\right] \\ =-\frac{1}{2}[cot{\left(\frac{\mathrm{\pi }}{2}\right)}^2-cot{\left(\frac{\mathrm{\pi }}{4}\right)}^2] \\ =\frac{1}{2}cot\left(\frac{{\mathrm{\pi }}^2}{16}\right)-\frac{1}{2}cot\left(\frac{{\mathrm{\pi }}^2}{4}\right) \end{gathered}$$

The exact value of the given definite integral is $\frac{1}{2}cot\left(\frac{{\mathrm{\pi }}^2}{16}\right)-\frac{1}{2}cot\left(\frac{{\mathrm{\pi }}^2}{4}\right)$.

Step 3. The graph to verify the answer is

The solution is area under graph which is

$$\begin{gathered} a\approx 1.330759 \\ \approx \frac{1}{2}\left[cot\right(\frac{{\mathrm{\pi }}^2}{16})-cot(\frac{{\mathrm{\pi }}^2}{4}\left)\right] \end{gathered}$$