Question

Evaluate each of the following integrals:

${\int }_0^{\frac{\mathrm{\pi }}{2}}{e}^x\left(\sin x-\cos x\right)dx$ [CBSE 2014]

Answer 1

Disclaimer: The solution has been provided by taking the lower limit of integral as 0.

$$\begin{gathered} {\int }_0^{\frac{\mathrm{\pi }}{2}}{e}^x\left(\sin x-\cos x\right)dx \\ =-{\int }_0^{\frac{\mathrm{\pi }}{2}}{e}^x\left[\cos x+\left(-\sin x\right)\right]dx \\ =-{\overline{)e^x\cos x}}_0^{\frac{\mathrm{\pi }}{2}}\left\{\int e^x\left[f\left(x\right)+f\text{'}\left(x\right)\right]dx=e^{x}f\left(x\right)+C\right\} \\ =-\left(e^{\frac{\mathrm{\pi }}{2}}\cos \frac{\mathrm{\pi }}{2}-e^0\cos 0\right) \\ =-\left(e^{\frac{\mathrm{\pi }}{2}}\times 0-1\times 1\right) \\ =-\left(0-1\right) \\ =1 \end{gathered}$$