Direct Substitution

Put y = sin x

Then

dy = cos x dx

Our function will be

f(y) = e^y

The limits of integration are 1 to -1

Then the answer is

e - 1/e

a = e

As a matter of fact both 'e' and '-1/e' can be the values of 'a' in the problem

d(exp(sin(x))) = exp(sin(x))cos(x)dx

Always remember, when there is "e" in the integral, the answer is always in the form of "e"

Let u = sinx Derivative of u = cosx Limits of integration u=1, u = -1 Integration of e power u is same Substitute upper limit and lower limit in e^u = e^1 -e^-1 Compare both left and right side values hence a = e

If we let $u=\sin x$ then $\frac{du}{dx}=\cos x$ . Substituting into the integral gets $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{du}{dx} e^{u} dx = \int_{-1}^{1} e^{u} du = \left(e^u + C\right)|_{-1}^{1} = e - \frac{1}{e}$ That means $a=e$ .

It can only be solved by"By Parts".

There are mainly 2 easy way i can see.. 1) Let, e^sinx =t 2) Use Leibniz rule

use substitution: $u = sin(x) \leftrightarrow du = cos(x) dx$

utilize: $\int e^u du = e^u + C$ and 2nd fundamental theorem of calculus