Advertisements
Advertisements
प्रश्न
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =
पर्याय
`(1)/(2)sqrt(x + 1) + c`
`(2)/(3)(x + 1)^(3/2) + c`
`sqrt(x + 1) + c`
`2(x - 1)^(3/2) + c`
Advertisements
उत्तर
`(2)/(3)(x + 1)^(3/2) + c`
Explanation:
`I = int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx`
I = `int sqrt((1+x)^2+ sqrtx *sqrt(1+ x) )/ (sqrt(x) + sqrt(1+x))*dx`
I = `int( sqrt(1 + x) sqrt(1 + x) + sqrtx )/ (sqrt(x)+sqrt(1 + x))*dx`
I `= int(sqrt(1+x)) dx = 2/3 (x + 1)^(3/2) + c`
I = `(2)/(3)(x + 1)^(3/2) + c`
APPEARS IN
संबंधित प्रश्न
Evaluate :`intxlogxdx`
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Find `intsqrtx/sqrt(a^3-x^3)dx`
Integrate the functions:
`(log x)^2/x`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`e^(2x+3)`
Integrate the functions:
`x/(e^(x^2))`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Integrate the following w.r.t. x : x3 + x2 – x + 1
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
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:
`(5 - 3x)(2 - 3x)^(-1/2)`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x : cos7x
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int 1/(sqrt"x" + "x")` 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(cos^2 x - sin^2 x))/(cos^2 x + sin^2 x)` dx = ______________
`int 1/(xsin^2(logx)) "d"x`
`int x/(x + 2) "d"x`
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
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 ______.
`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
Evaluate the following
`int1/(x^2 +4x-5)dx`
Evaluate `int(1 + x + x^2/(2!))dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
What is integration by substitution?
Which substitution is appropriate for \[\sqrt{\frac{x}{a-x}}\], \[\sqrt{\frac{a-x}{x}}\], \[\sqrt{x(a-x)}\], or \[\frac{1}{\sqrt{x(a-x)}}\]?
Integration by substitution is the reverse process of which rule?
