Integration with limits

Question

f(x) = (integration with limits 2x at the top -3 at the bottom) (2^(t[ |t+2|−(t+2)]))(4t + 4) dt.

Find / evaluate f(1) and derivative of f(x)

Answer 1

I see no way of integrating this integrand except by breaking it up into pieces.

To the left of -2, the integrand is (4x+4)*2^-(x+2) and to the right of -2 the integrand is 4x+4.

The area to the left of -1 will have a negative sign and the area to the right of -1 will have a positive sign.

The antiderivative to the left of -2 will require integration by parts.

The antiderivative to the right of -2 is easy...but remember the value will be a sum of negative and positive areas.

The derivative of f(x) should be obtained by the chain rule; it is the composition of the integrand with the function h(x)=2x.