Question

The value of${\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{2}}{e}^x\left(\log \left(\sin \right(x\left)\right)+\cot \left(x\right)\right)dx=$

• A. $e^{\frac{\mathrm{\pi }}{4}}\log \left(2\right)$
• B. $-e^{\frac{\mathrm{\pi }}{4}}\log \left(2\right)$
• C. $\left(\frac{1}{2}\right)e^{\frac{\mathrm{\pi }}{4}}\log \left(2\right)$
• D. $\left(-\frac{1}{2}\right)e^{\frac{\mathrm{\pi }}{4}}\log \left(2\right)$

Answer 1

The correct option is D

$\left(-\frac{1}{2}\right)e^{\frac{\mathrm{\pi }}{4}}\log \left(2\right)$

Explanation for the correct option:

Compute the required value:

Given:${\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{2}}{e}^x\left(\log \left(\sin \right(x\left)\right)+\cot \left(x\right)\right)dx$

Let $e^x\log \left(\sin \right(x\left)\right)=t$

$$\begin{gathered} \left(e^x\log \left(\sin \right(x\left)\right)+e^x\mathrm{co}t\left(x\right)\right)dx=dt \\ {e}^x\left(\log \left(\sin \right(x\left)\right)+\mathrm{co}t\left(x\right)\right)dx=dt \end{gathered}$$ (using product role of differentiation)

When $x=\frac{\mathrm{\pi }}{2},t=e^{\frac{\mathrm{\pi }}{2}}\log \left(\sin \left(\frac{\mathrm{\pi }}{2}\right)\right)\Rightarrow e^{\frac{\mathrm{\pi }}{2}}·\log \left(1\right)\Rightarrow 0$

$x=\frac{\mathrm{\pi }}{4},t=e^{\frac{\mathrm{\pi }}{4}}\log \left(\sin \left(\frac{\mathrm{\pi }}{4}\right)\right)\Rightarrow e^{\frac{\mathrm{\pi }}{4}}·\log \left(\frac{1}{\sqrt{2}}\right)\Rightarrow -\frac{1}{2}{e}^{\frac{\mathrm{\pi }}{4}}\log \left(2\right)$

$$\begin{gathered} {\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{2}}{e}^x\left(\log \left(\sin \right(x÷\left)\right)+\cot \left(x\right)\right)dx={\int }_0^{-\frac{1}{2}{e}^{\frac{\mathrm{\pi }}{4}}\log \left(2\right)}1dt \\ \Rightarrow {\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{2}}{e}^x\left(\log \left(\sin \right(x\left)\right)+\cot \left(x\right)\right)dx={\left|t\right|}_0^{-\frac{1}{2}{e}^{\frac{\mathrm{\pi }}{4}}\log \left(2\right)} \\ \Rightarrow {\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{2}}{e}^x\left(\log \left(\sin \right(x\left)\right)+\cot \left(x\right)\right)dx=-\frac{1}{2}{e}^{\frac{\mathrm{\pi }}{4}}\log \left(2\right) \end{gathered}$$

Hence, option $D$ is the correct answer.