Here we will use the Power Rule, which says that . First, let’s simplify the
integrand: = = . Now, we
can evaluate the integral: ∫(x^4 − 2x^2 + x−2) dx = = .
8.
Here we will use the Power Rule, which says that .
First, let’s simplify the integrand: ∫x(x − 1)^3 dx = ∫x(x^3 − 3x^2 + 3x − 1) dx = ∫(x^4 − 3x^3 + 3x^2 −
x) dx.
Now, we can evaluate the integral: ∫(x^4 − 3x^3 + 3x^2 − x) dx = =
.
- tan x + sec x + C
Here we will use the Rules for the Integrals of Trig Functions, namely: ∫sec^2 x dx = tan x + C
and ∫(sec x tan x) dx = sec x + C. First, let’s expand the integrand: ∫sec x (sec x + tan x) dx = ∫
(sec^2 x + sec x tan x) dx. We get ∫(sec^2 x + sec x tan x) dx = tan x + sec x + C.
- tan x + + C
Here we will use the Rules for the Integrals of Trig Functions, namely: ∫sec^2 x dx = tan x + C
and the Power Rule, which says that . We get ∫(sec^2 x + x) dx =
.
- sin x + 4 tan x + C
Here we will use the Rules for the Integrals of Trig Functions, namely: ∫sec^2 x dx = tan x + C
