Browse Questions

$\int\limits_0^{\Large\frac{\pi}{2}}\cos x\;e^{\sin x} dx$ is equal to ____________.

Toolbox:
• If $f(x)$ is substituted by $f(t)$,then $f'(x)dx=f'(t)dt$
• Hence $\int f(x)dx=\int f(t)dt.$
• $\int e^xdx=e^x+c$
Step 1:
Let $I=\int_0^{\Large\frac{\pi}{2}}\cos x e^{\sin x} dx.$
Put $\sin x=t$
$\cos xdx=dt.$
On differentiating with respect to t, we get as we substitute t, the limits also change.
When $x=0,t=\sin 0=0$
$x=\large\frac{\pi}{2},$$t=\sin\large\frac{\pi}{2}$$=1$
$I=\int_0^1e^t.dt.$
Step 2:
On integrating we get,
$I=[e^t]_0^1=e^1-e^0$
But $e^0=1$
Hence $I=e^1-1$
$\int_0^{\Large\frac{\pi}{2}}\cos x e^{\sin x} dx=e-1$