Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Advertisements
उत्तर
Let I = `int(1)/(sqrt(x) + sqrt(x^3)).dx`
= `int(1)/(x^(1/2)+ x^(3/2)).dx`
Put x = t2
∴ dx = 2t dt
Also `x^(1/2) = (t^2)^(1/2)` = t
and
`x^(3/2) = (t^2)^(3/2)` = t3
∴ I = `int (2tdt)/(t + t^3)`
= `2int "tdt"/(t(1 + t^2)`
= `2int (1)/(1 + t^2)dt`
= 2tan–1 t+ c
= `2tan^-1(sqrt(x)) + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`
Integrate the functions:
`(2x)/(1 + x^2)`
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
tan2(2x – 3)
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`1/(1 - tan x)`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Integrate the following function w.r.t. x:
`(10x^9 +10^x.log10)/(10^x + x^10)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
`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 = ?
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
`int ("e"^x(x - 1))/(x^2) "d"x` = ______
`int logx/x "d"x`
`int sqrt(x) sec(x)^(3/2) tan(x)^(3/2)"d"x`
`int x/(x + 2) "d"x`
`int(1 - x)^(-2) dx` = ______.
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
Evaluate `int(1+ x + x^2/(2!)) dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Solve the following Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3)dx`
Evaluate `int (1)/(x(x - 1))dx`
Evaluate.
`int (5x^2 -6x + 3)/(2x -3)dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
