Evaluate the integral: x^2 ln(x+7) dx from limit 0 to 1 (0,1) I'm stuck!!!?

Question

Ok, so by using Integration by Parts theory: uv dx = u I(v) - Integral of D(u) I(v) dx,

I chose to use the derivative of ln(x+7), which is: 1/(x+7); integration of x^2, which is x^3/3

so I write the problem x^2 ln(x+7) dx Limit (0,1) to:

ln(x+7) x^2 dx

=ln(x+7) times x^3/3 - integral of 1/(x+7) times x^3/3 dx


then I have no clue where to go from there, 5 answers are:

a. (344 ln8 - 343 ln7) / 3 -(275/18)
b. (344 ln8 + 343 ln7) / 3 - (275/9)
c. (344 ln8 - 343 ln7) / 3 - (401/9)
d. (344 ln8 - 343 ln7) / 3 - (401/18)
e. (344 ln8 + 343 ln7) / 3 + (401/9)

I'm already stuck in the beginning and have no idea how the correct answer ends with 344 ln8 etc........someone help please, thank you!

Thanks for the help, but is there another way to approach this problem since I don't know how to do the short/long division for integrals.

Answer 1

Integral (0 to 1, x^2 ln(x + 7) dx )

Use integration by parts.
Let u = ln(x + 7). dv = x^2 dx
du = 1/(x + 7) dx. v = (1/3)x^3

(1/3)x^3 ln(x + 7) - Integral ( (1/3)x^3 (1/(x + 7)) dx )

First, pull out the constant (1/3) from the integral.

(1/3)x^3 ln(x + 7) - (1/3) * Integral ( x^3 (1/(x + 7)) dx )

Now, simplify.

(1/3)x^3 ln(x + 7) - (1/3) * Integral ( x^3/(x + 7) dx )

Use long division on the integral; (x + 7) into x^3.
Since synthetic long division is difficult to show on here, I'm going to just tell you the quotient and remainder are x^2 - 7x + 49 and -343 respectively, meaning
x^3/(x + 7) = -343/(x + 7) + x^2 - 7x + 49

(1/3)x^3 ln(x + 7) - (1/3) * Integral ( [-343/(x + 7) + x^2 - 7x + 49] dx )

Which we can evaluate directly (barring minor steps)

(1/3)x^3 ln(x + 7) - (1/3) * [ -343ln|x + 7| + (1/3)x^3 - (7/2)x^2 + 49x ] {evaluated from 0 to 1}

(1/3)x^3 ln(x + 7) + (343/3) ln|x + 7| - (1/9)x^3 + (7/6)x^2 - (49/3)x {evaluated from 0 to 1}

[ (1/3) ln(8) + (343/3) ln|8| - (1/9) + (7/6) - (49/3) ] -
[ 0 + (343/3)ln(7) - 0 + 0 - 0]

(1/3) ln(8) + (343/3) ln(8) - (1/9) + (7/6) - (49/3) - (343/3)ln(7)

You take it from here.

Answer 2

Integrate by parts
=x^3/3*ln(x+7) -1/3 Int (x^3/(x+7) dx
So we have to evaluate Int(x^3/(x+7)dx
Firs you have to do the division
x^3/(x+7) = x^2-7x+49 -343/(x+7) (by short division)
so the integral is x^3/3-7/2x^2+49x-343 lnIx+7I
So we have
1/3[x^3 ln(x+7)-(x^3/3-7/2x^2+49x -343 lnIx+7I] between 0 and 1
=1/3[ln8 -1/3+7/2-49+343 ln8-343 ln 7] =
(344 ln8 -343ln7)/3 -275/18
so the answer is (a)