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Step-by-step explanation:

what the heck lol

480 Rods

1320 Fathoms

(A) true.

(B). False.

Step-by-step explanation:

(A). F(x) = 4xe^-2x.

Let's make the assumption that 2x = b (1).

Therefore, taking the differentiation with respect to x, we have;

2x dx = db.

The next thing thing to do is to integrate, taking the upper limit to be infinity and the lower limit to be zero:

∫( b e^-b db).

Changing the Lim. infinity = 2.

= ∫( b e^-b db). (2).

The step (2) above can be solve with Integrations by part;

Lim. 2 ---> infinity [ b e^-b + ∫( 1 e^-b db|

( | = Upper boundary = 2 and the lower boundary = 0).

Lim. 2 > infinity [ 2 e^-2 - 0] - lim. 2 --> infinity [ 0 - 1].

Lim 2 ---> infinity [ 2/e^2 + 1].

Let 2/e^2 = j.

The, lim 2 ---> infinity j + 1.

To solve this, there is need to make use of L'hospital rule,

j = Lim 2 ---> infinity [ 1 /e^2] = 0.

Thus, j = 0. And 0 + 1 = 1.

(PROVED TO BE TRUE).

(B). Taking limit from 1 (upper) to 0( lower). Assuming that b = 3 - 4x.

Therefore, -4dx = db.

∫( 8 /(3 - 4x)^3 dx.

Taking the upper boundary = -1 and the lower boundary = 3.

∫ ( - 2db/ b^3.

= 2 ∫ ( - db/ b^3.

= 2 [ b^-3 + 2) / -3 + 1.

= 2 | - 1/ 2 db

= 8/9.

Not true.