Question

Evaluate each of the following integral:

\[\int_0^\frac{\pi}{2} e^x \left( \sin x - \cos x \right)dx\]

 

Answer 1

\[\int_0^\frac{\pi}{2} e^x \left( \sin x - \cos x \right)dx\]
\[ = - \int_0^\frac{\pi}{2} e^x \left[ \cos x + \left( - \sin x \right) \right]dx\]
\[ = \left.- {e^x \cos x}\right|_0^\frac{\pi}{2} .............\left\{ \int e^x \left[ f\left( x \right) + f'\left( x \right) \right]dx = e^x f\left( x \right) + C \right\}\]
\[ = - \left( e^\frac{\pi}{2} \cos\frac{\pi}{2} - e^0 \cos0 \right)\]
\[ = - \left( e^\frac{\pi}{2} \times 0 - 1 \times 1 \right)\]
\[ = - \left( 0 - 1 \right)\]
\[ = 1\]