Question

Need help to evaluate integral? 

Answer 1

Given function  ₄∫⁶ ( t + 9 ) ( t – 2 )⁶

Apply formula ∫UVdx = U∫Vdx - ∫(dU/dx)(∫Vdx)dx

=  (t + 9) ∫ (t – 2)⁶ - ∫ (d/dt) (t + 9) ∫ (t – 2)⁶ dt

= (1/7)(t + 9) (t – 2)⁷ - ∫ (1) (1/7) (t – 2)⁷ dt

= (1/7)(t + 9) (t – 2)⁷ - (1/7)∫ (t – 2)⁷ dt

= (1/7)(t + 9) (t – 2)⁷ - (1/7)(1/8) (t – 2)⁸

= (1/7)(t + 9) (t – 2)⁷ - (1/56) (t – 2)⁸

Apply limits  (  lower limit = 4 , upper limit = 6  )

= (1/7) [ (6 + 9) (6 – 2)⁷ - (4 + 9) (4 – 2)⁷ ] - (1/56) [ (6 – 2)⁸ - (4 – 2)⁸ ]

= (1/7) [ (15)(4)⁷ - (13)(2)⁷ ] - (1/56) [(4)⁸ - (2)⁸]

= (1/7) [ (15) (16384) - (13) (128) ] - (1/56) [ (65536) - (256) ]

= (1/7) [ 245760 - 1664 ] - (1/56) [ 65280 ]

= (1/7) [ 244096 ] - (1/7) [ 8160 ]

= (1/7) (244096 – 8160)

= (1/7) (235936)

= 235936/7

Solution : ₄∫⁶ ( t + 9 ) ( t – 2 )⁶ = 235936/7.