State whether the following statement is True or False. The proper substitution for ∫x(xx)x(2logx+1) dx is (xx)x = t - Mathematics and Statistics

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MCQ
True or False

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

  Is there an error in this question or solution?
Chapter 1.5: Integration - Q.3

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