The following integral requires a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate the following integral. x^4 + 7/x^3 + 2x dx Find the partial fraction decomposition of the integrand. x^4 + 7/x^3 + 2x dx

Division yields

Now for partial fractions: you're looking for constants a, b, and c such that

which gives a + b = 2, c = 0, and 2a = -7, so that a = -7/2 and b = 11/2. Then

Now, in the integral we get

The first two terms are trivial to integrate. For the third, substitute y = x ² + 2 and dy = 2x dx to get

-8

step-by-step explanation:

g = (2, 2)

step-by-step explanation:

given the ratio is 1 : 1 then f is the midpoint of eg

let the coordinates of g = (x, y)

using the midpoint formula

(0 + x) = 1 ( multiply both sides by 2 )

0 + x = 2 ⇒ x = 2

similarly

(4 + y) = 3 ( multiply both sides by 2 )

4 + y = 6 ( subtract 4 from both sides )

y = 2

coordinates of g = (2, 2)