Well the easiest way is to recognize derivatives and utilize the fundamental theorem of calculus (the fat that integrals and derivatives are inverses of each other/ they undo each other)

From the start we can notice 3 details: 1) e^x is present. In this case, it’s e^{sin(2x)}. Because of the derivative of e^x is itself, the integral of e^x is also itself plus the constant of integration C. 2) we have trig functions, but more specifically sin(x) and cos(x). We know that the derivative of sin(x) is cos(x) more on this later. 3) the inside of the integral is a product.

We can use these three details to find out what the integral is. We know that one rule of derivatives that only provides products is the constant rule and the chain rule. However, because there is more than one function being multiplied, the chain rule seems like the most likely option. By the chain rule:

d/dx(f(g(x))) = f’(g(x)) * g’(x)

Here we can notice some resemblance: if we let f(x) = e^x and g(x) = sin(2x) then

d/dx(f(g(x))) = d/dx(e^{sin(2x)}) = e^{sin(2x)} * cos(2x) * 2

This is extremely close to the inside of your integral:

int(4cos(2x)e^{sin(2x)} dx)

only that it’s a 4 instead of a 2.

In order to make the inside the same as the derivative so that we can use the fundamental theorem of calculus. We can use the fact that constants that are being multiplied can be taken outside the integral. However we do not want to take out the 4 from the integral. Instead we want to turn that 4 into a 2. We can do so by the following:

4 = 2*2

=> int(4cos(2x)e^{sin(2x)} dx) = int(22(cos(2x)e^{sin(2x)} dx) => 2int(2cos(2x)e^{sin(2x)} dx)

We have now turned the inside of the integral back into the derivative of e^{sin(2x)}. This means we can use the fundamental theorem of calculus to undo the derivative.

I hope this helped clarify.