Question

Find the intervals in `R` over which `∫(t+1)^3 e^t dt` from `-1` to `x` is decreasing?

Answer 1

`f(x) = int_(-1)^x (t+1)^3e^tdt`

is decreasing for `x in (-oo,-1)`

Explanation:

Given:

`f(x) = int_(-1)^x (t+1)^3e^tdt`

base on the fundamental theorem of calculus:

`(df)/dx = (x+1)^3e^x`

Now, the intervals where `f(x)` is decreasing are the intervals where its derivative is negative, so we must solve the inequality:

`(df)/dx < 0`

`(x+1)^3e^x < 0`

as `e^x` is always positive:

`(x+1)^3 < 0`

and as the cubic root preserves the sign:

`(x+1) < 0`

that is:

`x < -1`

Thus `f(x)` is decreasing for `x < -1`