Advertisements
Advertisements
प्रश्न
उत्तर
\[\int\left( \frac{x - 1}{x^2} \right) e^x dx = \int\left( \frac{x}{x^2} - \frac{1}{x^2} \right) e^x dx\]
\[ = \int\left( \frac{1}{x} - \frac{1}{x^2} \right) e^x dx\]
\[\text{ Consider,} f\left( x \right) = \frac{1}{x},\text{ then f}^ \left( x \right) = - \frac{1}{x^2}\]
\[\text{ Thus , the given integrand is of the form e}^x \left[ f\left( x \right) + f^ \left( x \right) \right] . \]
\[\text{ Therefore, }\int\left( \frac{x - 1}{x^2} \right) e^x dx = \frac{1}{x} e^x + C\]
\[\text{ Hence,} f\left( x \right) = \frac{1}{x} .\]
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`1/(1 - tan x)`
`int (dx)/(sin^2 x cos^2 x)` equals:
Write a value of
Write a value of
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: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`
Integrate the following functions w.r.t. x:
`(10x^9 10^x.log10)/(10^x + x^10)`
Evaluate the following : `int sinx/(sin 3x).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Choose the correct options from the given alternatives :
`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/(4"x"^2 - 20"x" + 17)` dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3)dx`
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` dx
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int cos^7 x "d"x`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
If f'(x) = `x + 1/x`, then f(x) is ______.
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
If f'(x) = 4x3- 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate.
`int (5x^2 -6x + 3)/(2x -3)dx`
Evaluate:
`intsqrt(sec x/2 - 1)dx`
