Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Advertisements
उत्तर
Let I = e3logx(x4 + 1)–1.dx
= `int e^(logx^3)/(x^4 + 1).dx`
= `int x^3/(x^4 + 1).dx` ...[∵ elogN = N]
= `(1)/(4) int(4x^3)/(x^4 + 1).dx`
= `(1)/(4) int(d/dx(x^4 + 1))/(x^4 + 1).dx`
= `(1)/(4)log|x^4 + 1| + c. ...[∵ int (f'(x))/f(x) dx = log|f(x)| + c]`
APPEARS IN
संबंधित प्रश्न
Evaluate :
`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Integrate the functions:
`x/(9 - 4x^2)`
Integrate the functions:
`e^(tan^(-1)x)/(1+x^2)`
Integrate the functions:
`(sin^(-1) x)/(sqrt(1-x^2))`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Integrate the functions:
`(1+ log x)^2/x`
`(10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx` equals:
Evaluate : `∫1/(3+2sinx+cosx)dx`
Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`
Write a value of
Write a value of
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int \log_e x\ dx\].
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
Evaluate the following integrals:
`int x/(x + 2).dx`
Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Choose the correct options from the given alternatives :
`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =
If f'(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
Evaluate `int (5"x" + 1)^(4/9)` dx
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int(1 - x)^(-2) dx` = ______.
`int (1 + x)/(x + "e"^(-x)) "d"x`
`int sec^6 x tan x "d"x` = ______.
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.
Evaluate `int (1+x+x^2/(2!))dx`
Evaluate the following.
`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate:
`intsqrt(3 + 4x - 4x^2) dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Which standard substitution is used for \[\sqrt{\mathrm{a}^2-x^2}\], \[\frac{1}{\sqrt{\mathrm{a}^2-x^2}}\], or \[\mathrm{a}^2-x^2\]?
For \[\sqrt{\frac{x-\alpha}{\beta-x}}\] or \[\sqrt{(x-\alpha)(\beta-x)}\], where \[\beta>\alpha\], which substitution is used?
