Can I antidifferentiate this?

Question

Can i antidifferentiate x/[(x+4)^3] using "basic" level of mathematics? (unsure of the course name as it can differ from country to country).

Thanks a lot.

Answer 1

Oh yes. You can!

write the Nr as x+4 - 4 so you want to find the antiderivative of

[1/(x+4)^2] - [4/(x+4)^3]

i.e that of (x+4)^(- 2) - 4(x+4)^( - 3)

= - 1/(x+4) + 2(x+4)^(-2)+c

Answer 2

∫ [x / (x + 4)³] dx =

since the denominator is already factored, you can decompose it straight

into partial fractions as:

x / (x + 4)³ = A/(x + 4) + B/(x + 4)² + C/(x + 4)³ →

find out the LCD at right side:

x / (x + 4)³ = [A(x + 4)² + B(x + 4) + C] / (x + 4)³ →

x / (x + 4)³ = [A(x² + 8x + 16) + B(x + 4) + C] / (x + 4)³ →

x / (x + 4)³ = (Ax² + 8Ax + 16A + Bx + 4B + C) / (x + 4)³ →

x / (x + 4)³ = [Ax² + (8A + B)x + (16A + 4B + C)] / (x + 4)³ →

equating the numerators you get the system:

| A = 0

| 8A + B = 1 → 0 + B = 1 → B = 1

| 16A + 4B + C = 0 → 0 + 4(1) + C = 0 → C = - 4

therefore (see above):

x / (x + 4)³ = A/(x + 4) + B/(x + 4)² + C/(x + 4)³ →

x / (x + 4)³ = 0/(x + 4) + 1/(x + 4)² - 4/(x + 4)³ →

x / (x + 4)³ = 1/(x + 4)² - 4/(x + 4)³

thus:

∫ [x / (x + 4)³] dx = ∫ {[1/(x + 4)²] - [4/(x + 4)³]} dx =

breaking it up and factoring the constant out,

∫ [1/(x + 4)²] dx - 4 ∫  [1/(x + 4)³] dx =

rewrite the integrand as negative powers:

∫ (x + 4)^(-2) dx - 4 ∫ (x + 4)^(-3) dx =

∫ (x + 4)^(-2) d(x + 4) - 4 ∫ (x + 4)^(-3) d(x + 4) =

[(x + 4)^(-2+1)] /(-2+1) - 4 [(x + 4)^(-3+1)] /(-3+1) + C =

[(x + 4)^(-1)] /(-1) - 4 [(x + 4)^(-2)] /(-2) + C =

[- 1/(x + 4)] - 4(-1/2) [1/(x + 4)²] + C =

[- 1/(x + 4)] + 2 [1/(x + 4)²] + C

thus, in conclusion,

∫ [x / (x + 4)³] dx = [- 1/(x + 4)] + 2 [1/(x + 4)²] + C

I hope it helps...

Bye!

Answer 3

In the USA, we call this "integrate".  Yes, it isn't that hard to do.

Using u-substitution:

u = x+4 (and x = u-4)

so du = dx

Substitute that into your original formula:

∫x/[(x+4)^3] dx = ∫(u-4)/(u^3) du

=∫u/(u^3)du - ∫4/(u^3)du

=∫u^(-2)du - ∫4u^(-3)du

= -u^(-1) + 2u^(-2) + c

= -1/u + 2/(u^2) + c

= -1/(x+4) + 2/[(x+4)^2] + c