Question

How do you do this question?

Answer 1

Answer:

18 ln(9) − ¹⁰⁴/₉

Step-by-step explanation:

∫₁⁹ √y ln(y) dy

If u = ln(y), then du = dy/y.

If dv = √y dy, then v = ⅔ (y)^³/₂.

∫ u dv = uv − ∫ v du

= ln(y) (⅔ (y)^³/₂) − ∫ (⅔ (y)^³/₂) (dy/y)

= ln(y) (⅔ (y)^³/₂) − ∫ (⅔ (y)^½) dy

= ln(y) (⅔ (y)^³/₂) − ⁴/₉ (y)^³/₂ + C

= (⅔ (y)^³/₂) (ln(y) − ⅔) + C

Evaluate between y=1 and y=9.

[(⅔ (9)^³/₂) (ln(9) − ⅔)] − [(⅔ (1)^³/₂) (ln(1) − ⅔)]

[18 (ln(9) − ⅔)] − [(⅔) (-⅔)]

18 ln(9) − 12 + ⁴/₉

18 ln(9) − ¹⁰⁴/₉