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Question
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
Solution
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x5 + `underline((-5)/3)` x3 + 5x + c
Explanation:
`"I" = int (5(x^6 + 1))/(x^2 + 1) "dx"`
`"I" = 5 int ((x^2)^3 + (1)^3)/("x"^2 + 1) "dx"`
`"I" = 5int((cancel("x"^2 + 1))("x"^4 - "x"^2 + 1))/(cancel("x"^2 + 1)) "dx" ...[a^3 + b^3 = (a + b)(a^2 - ab + b^2)]`
`"I" = 5 int ("x"^4 - "x"^2 + 1)` dx
`"I" = 5 ("x"^5/5 - "x"^3/3 + "x") + c ...[int "x"^"n" "dx" = "x"^("n" + 1)/("n" + 1)]`
`"I" = "x"^5 - 5/3"x"^3 + 5"x"` + c
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