Question

Evaluate $\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\sin x-\cos x)dx.$

Answer 1

Hint: The given integration is of type$\int{{{e}^{x}}\left[ f(x)+{{f}^{'}}(x) \right]}dx$.
we can use $\int{{{e}^{x}}\left[ f(x)+{{f}^{'}}(x) \right]}dx$$={{e}^{x}}.f(x)+c$, where c is constant. So first we find integration by using the property and after that we put limits.

Complete step-by-step answer:
Now we have to evaluate $\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\sin x-\cos x)dx$
$\Rightarrow -$ $\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\cos x-\sin x)dx$
Here we take – sign outside of the integral sign. Now suppose $f(x)=\cos x$ so we can write ${{f}^{'}}(x)=-\sin x$
So we have$\int{{{e}^{x}}\left[ f(x)+{{f}^{'}}(x) \right]}dx$$={{e}^{x}}.f(x)+c$
$\Rightarrow -$ $\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\cos x+(-\sin x))dx$=$\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(f(x)+f'(x))dx$
$\Rightarrow -$ $\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\cos x-\sin x)dx$=$\left[ {{e}^{x}}f(x) \right]_{0}^{\dfrac{\pi }{2}}$
Putting value of f(x) =cosx we can write
$\Rightarrow -$ $\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\cos x-\sin x)dx$=$\left[ {{e}^{x}}\cos x \right]_{0}^{\dfrac{\pi }{2}}$

Now substituting lower and upper limits we have
$\Rightarrow -$ $\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\cos x-\sin x)dx$=$\left[ {{e}^{\dfrac{\pi }{2}}}\cos \dfrac{\pi }{2}-{{e}^{0}}\cos 0 \right]$
As we know $\cos \dfrac{\pi }{2}=0;{{e}^{0}}=1;and\text{ cos0=1}$
So we can write
$\Rightarrow -$ $\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\cos x-\sin x)dx$=$\left[ {{e}^{\dfrac{\pi }{2}}}\times 0-{{e}^{0}}\times 1 \right]$
$\Rightarrow -$ $\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\cos x-\sin x)dx$=$-1$
$\Rightarrow \int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\cos x-\sin x)dx=1$
Hence
$\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\cos x-\sin x)dx=1$

Note: This integration can be solved by integrating $\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\cos x)dx-\int\limits_{0}^{\dfrac{\pi }{2}}{\sin x}dx$ separately by using integration by parts for$\int\limits_{0}^{\dfrac{\pi }{2}}{{{e}^{x}}}(\cos x)dx$