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Question
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)dx
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Solution
Let I = ∫ (x + 1)(x + 2)7 (x + 3)dx
Put x + 2 = t
∴ dx = dt
Also, x = t - 2
∴ x + 1 = t - 2 + 1
= t - 1
and x + 3 = t - 2 + 3
= t + 1
∴ I = `int ("t" - 1) * "t"^7 ("t" + 1) * "dt"`
`= int ("t"^2 - 1) * "t"^7 * "dt"`
`= int ("t"^9 - "t"^7) "dt"`
`= int "t"^9 "dt" - int "t"^7 "dt"`
`= "t"^10/10 - "t"^8/8 + "c"`
∴ I = `("x + 2")^10/10 - ("x + 2")^8/8` + c
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