Here's (c)
5∫ x / √(x + 1) dx .....let's integrate by "parts".....we'll go back and account for the "5" later on...
Let u = x so... du = dx
Let dv = 1 / √(x + 1) and ..... v = 2 √(x + 1) .... so we have
∫ u dv = u*v - ∫ v dv = 2x √(x + 1) - ∫ 2 √(x + 1) dx = 2 [x √(x + 1) - ∫ √(x + 1) dx ]
∫ u dv = 2[x √(x + 1) - ∫ √(x + 1) dx ]
And, substituting in the limits for for the first term we have
(2)[3 √(3 + 1) - 1√(1 + 1)] = (2)[6 - √(2) ]
Now, for the second integral, let u = x + 1 du = dx ... so we have....
-2∫ u^(1/2) du =
(-2)(2/3) u^(3/2) =
(-4/3)(x + 1)^(3/2)
And substituting in the limits of integration, we have
(-4/3)[(3 + 1)^(3/2) - (1 + 1)^(3/2)] =
(4/3) [2^(3/2) - 4^(3/2)] =
(4/3)[ √8 - 8]
Putting everything together, we have
(2)(6 - √2 ) + (4/3) (√8 - 8 ) = about 2.2761423749153967
But remenber, in the original integrlal, we have 5 times this much ....so we have about ..... 11.3807118745769835
