Advertisements
Advertisements
प्रश्न
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
विकल्प
True
False
उत्तर
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
APPEARS IN
संबंधित प्रश्न
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Evaluate: `int 1/(x(x-1)) dx`
Write a value of
Write a value of
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
`int logx/x "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3) dx`
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
Evaluate `int 1/(x(x-1)) dx`
