Advertisements
Advertisements
प्रश्न
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
उत्तर
Let I = `int sqrt((10 + x)/(10 - x)).dx`
= `int sqrt((10 + x)/(10 - x) xx (10 + x)/(10 + x)).dx`
= `int (10 + x)/sqrt(100 - x^2).dx`
= `int (10)/sqrt(100 - x^2).dx + int x/sqrt(100 - x^2).dx`
= `10 int (1)/sqrt(10^2 - x^2).dx + (1)/(2) int (2x)/sqrt(100 - x^2).dx`
= I1 + I2 ...(Let)
I1 = `10 int (1)/sqrt(10^2 - x^2).dx`
= `10 sin^-1 (x/10) + c_1`
In I2, put 100 – x2 = t
∴ – 2x dx = dt
∴ 2x dx = – dt
I2 = `-(1)/(2) int t^(-1/2) dt`
= `-(1)/(2).t^(1/2)/((1/2)) + c_2`
= `- sqrt(100 - x^2) + c_2`
I = `10 sin^-1 (x/10) - sqrt(100 - x^2) + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate: `int sqrt(tanx)/(sinxcosx) dx`
Integrate the functions:
`1/(x + x log x)`
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Evaluate: `int 1/(x(x-1)) dx`
Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`
Write a value of
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int \log_e x\ dx\].
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
The value of \[\int\frac{1}{x + x \log x} dx\] is
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`
Evaluate the following integrals : tan2x dx
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`
Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`
Evaluate the following : `int (1)/(4x^2 - 3).dx`
Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`
Evaluate the following : `int (1)/(x^2 + 8x + 12).dx`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Evaluate `int 1/("x" ("x" - 1))` dx
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
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 1/(7 + 6"x" - "x"^2)` dx
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
State whether the following statement is True or False.
If ∫ x f(x) dx = `("f"("x"))/2`, then find f(x) = `"e"^("x"^2)`
Evaluate `int 1/((2"x" + 3))` dx
Evaluate: `int "e"^sqrt"x"` dx
`int sqrt(x^2 + 2x + 5)` dx = ______________
`int ("e"^(2x) + "e"^(-2x))/("e"^x) "d"x`
`int x^x (1 + logx) "d"x`
`int cot^2x "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 ______
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
`int(sin2x)/(5sin^2x+3cos^2x) dx=` ______.
`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.
`int sqrt(x^2 - a^2)/x dx` = ______.
`int (logx)^2/x dx` = ______.
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Evaluate `int1/(x(x - 1))dx`
Solve the following Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3)dx`
Evaluate the following.
`int x sqrt(1 + x^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(sqrt(tanx) + sqrt(cotx))dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate:
`int sin^3x cos^3x dx`
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
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 `int (5x^2 - 6x + 3)/(2x - 3) dx`
