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Question
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
Options
True
False
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Solution
True
Explanation:
Let I = ∫ x (xx)x (2 log x + 1) dx
Put `("x"^"x")^"x"` = t
Taking logarithm of both sides, we get
log `("x"^"x")^"x"` = log t
∴ `"x"^2 * log "x" = log "t"`
Differentiating w.r.t. x, we get
`"x"^2 * 1/"x" + (log "x") * "2x" = 1/"t" * "dt"/"dx"`
∴ `("x" + 2"x" log "x") "dx" = 1/"t" * "dt"`
∴ x(1 + 2 log x) dx = `1/"t" * "dt"`
∴ I = `int "t" * 1/"t" * "dt" = int 1 * "dt" = "t" + "c" = ("x"^"x")^"x"` + c
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