Advertisements
Advertisements
प्रश्न
उत्तर
\[\int \sqrt{x^2 + x + 1} \text{ dx}\]
\[ = \int \sqrt{x^2 + x + \left( \frac{1}{2} \right)^2 - \left( \frac{1}{2} \right)^2 + 1} \text{ dx}\]
\[ = \int \sqrt{\left( x + \frac{1}{2} \right)^2 + \left( \frac{\sqrt{3}}{2} \right)^2}\]
\[ = \left( \frac{x + \frac{1}{2}}{2} \right) \sqrt{\left( x + \frac{1}{2} \right)^2 + \left( \frac{\sqrt{3}}{2} \right)^2} + \frac{3}{8}\text{ log } \left| \left( x + \frac{1}{2} \right) + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| + C \left[ \because \int\sqrt{x^2 + a^2}dx = \frac{1}{2}x\sqrt{x^2 + a^2} + \frac{1}{2} a^2 \text{ ln }\left| x + \sqrt{x^2 + a^2} \right| + C \right]\]
\[ = \left( \frac{2x + 1}{4} \right) \sqrt{x^2 + x + 1} + \frac{3}{8}\text{ log }\left| \left( 2x + 1 \right) + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| + C\]
APPEARS IN
संबंधित प्रश्न
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Integrate the functions:
`(log x)^2/x`
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`cos x /(sqrt(1+sinx))`
Evaluate: `int 1/(x(x-1)) dx`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Write a value of
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`
Evaluate the following integrals : `int sin x/cos^2x dx`
Evaluate the following integrals : `int sin 4x cos 3x dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
`int ("x + 2")/(2"x"^2 + 6"x" + 5)"dx" = "p" int (4"x" + 6)/(2"x"^2 + 6"x" + 5) "dx" + 1/2 int "dx"/(2"x"^2 + 6"x" + 5)`, then p = ?
Evaluate `int 1/((2"x" + 3))` dx
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int sqrt(1 + sin2x) "d"x`
Evaluate `int"e"^x (1/x - 1/x^2) "d"x`
`int dx/(1 + e^-x)` = ______
`int sec^6 x tan x "d"x` = ______.
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
If f'(x) = `x + 1/x`, then f(x) is ______.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
Evaluate:
`int(sqrt(tanx) + sqrt(cotx))dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
