How do you find this definite integral?

Hello,

let's find the antiderivative, first;

∫ x² √(8x + 5) dx =

let:

√(8x + 5) = u

8x + 5 = u²

8x = u² - 5

x = (1/8)(u² - 5)

dx = (1/8)2u du

dx = (1/4)u du

then, substituting:

∫ x² √(8x + 5) dx = ∫ [(1/8)(u² - 5)]² u (1/4)u du =

(expanding)

∫ (1/64)(u^4 - 10u² + 25) (1/4)u² du =

∫ (1/256)(u^6 - 10u^4 + 25u²) du =

∫ [(1/256)u^6 - (5/128)u^4 + (25/256)u²] du =

let's split this pulling constants out:

(1/256) ∫ u^6 du - (5/128) ∫ u^4 du + (25/256) ∫ u² du =

(1/256) [1/(6+1)] u^(6+1) - (5/128) [1/(4+1)] u^(4+1) + (25/256) [1/(2+1)] u^(2+1) + C =

(1/256)(1/7)u^7 - (5/128)(1/5)u^5 + (25/256)(1/3)u³ + C =

(1/1792)u^7 - (1/128)u^5 + (25/768)u³ + C

let's substitute back √(8x + 5) for u, yielding:

(1/1792)√(8x + 5)^7 - (1/128)√(8x + 5)^5 + (25/768)√(8x + 5)³ + C

that is the antiderivative;

finally, having the antiderivative, let's plug in the bounds:

1

∫ x² √(8x + 5) dx = {(1/1792)√[8(1) + 5]^7 - (1/128)√[8(1) + 5]^5 + (25/768)√[8(1) +

0

5]^7 - (1/128)√[8(0) + 5]^5 + (25/768)√[8(0) + 5]³} =

(1/1792)√(8 + 5)^7 - (1/128)√(8 + 5)^5 + (25/768)√(8 + 5)³ - (1/1792)√5^7 +

(1/128)√5^5 - (25/768)√5³ =

(1/1792)√13^7 - (1/128)√13^5 + (25/768)√13³ - (1/1792)(5³)√5 + (1/128)(5²)√5 -

(25/768)5√5 =

(1/1792)(13³)√13 - (1/128)(13²)√13 + (25/768)13√13 - (125/1792)√5 + (25/128)√5 -

(125/768)√5 =

(2197/1792)√13 - (169/128)√13 + (325/768)√13 - (125/1792)√5 + (25/128)√5 -

(125/768)√5 =

{[(3)2197 - (42)169 + (7)325]/5376}√13 + {[- (3)125 + (42)25 - (7)125]/5376}√5 =

[(6591 - 7098 + 2275)/5376]√13 + [(- 375 + 1050 - 875)/5376]√5 =

(1768/5376)√13 - (200/5376)√5 =

(221/672)√13 - (25/672)√5 =

(1/672)(221√13 - 25√5)

the answer is: (1/672)(221√13 - 25√5) (≈1.10257)

I hope it helps..