Find the indefinite integral by making a change of variables. (Use C for the constant of integration.) (x + 5) 3 − x dx

Question

Find the indefinite integral by making a change of variables. (Use C for the constant of integration.)

(x + 5)3 − x dx

Answer 1

You could let u = 3 - x which would mean x = 3 - u and, taking the derivative we would get:

du/dx = -1

du = -dx

dx = -du or dx = (-1)du

Then, plugging in "u" for "3 - x" you get:

∫(x + 5)√u dx

And plugging in "3 - u" for "x" you get:

∫(3 - u + 5)√u dx

And by plugging in "(-1) du" for dx you get:

∫(3 - u + 5)√u (-1) du

Then simplifying:

∫-(8 - u)√u du

∫(u - 8)√u du

∫u√u - 8√u du

∫u√u du - ∫8√u du

∫u^3/2du - 8∫u^1/2du

(2/5)u^5/2 - 8(2/3)u^3/2 + C

(2/5)u^5/2 - (16/3)u^3/2 + C

Then back substitute "3 - x" in place of "u":

(2/5)(3 - x)^5/2 - (16/3)(3 - x)^3/2 + C

And there are ways to "simplify" of course if you don't like this particular form