Q229 / 270 · NDA 2018 (Phase 1) · Mathematics
What is $\int_0^\pi \mathrm{e}^{\mathrm{x}} \sin \mathrm{x} d \mathrm{x}$ equal to?
- Correct answerA$\frac{\mathrm{e}^\pi+1}{2}$
- B$\frac{\mathrm{e}^\pi-1}{2}$
- C$\mathrm{e}^\pi+1$
- D$\frac{\mathrm{e}^\pi+1}{4}$
Solution
$\begin{aligned} & \int_0^\pi e^x \sin x \cdot d x=I \\ & I=\left(\sin x \cdot e^x\right)_0^\pi-\int_0^\pi \cos x \cdot e^x \cdot d x \\ & I=\left(\sin \pi \cdot e^\pi-\sin 0 \cdot e^0\right)-\left\{\left[\cos x \cdot e^x\right]_0^\pi-\int_0^\pi \sin x \cdot e^x \cdot d x\right\} \\ & I=0-\left\{\left(\cos \pi \cdot e^\pi-\cos 0 \cdot e^0\right)-I\right\} \\ & I=-\left[-e^\pi-1\right]-I \\ & \Rightarrow 2 I=e^\pi+1 \Rightarrow I=\frac{e^\pi+1}{2}\end{aligned}$
Hence, the correct answer is option: 'A'.