Advertisements
Advertisements
प्रश्न
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
Advertisements
उत्तर
Let I = `int sqrt((x - 7)/(x - 9)).dx`
= `int sqrt((x - 7)/(x - 9) xx (x - 7)/(x - 7)).dx`
= `int sqrt((x - 7)^2/(x^2 - 16x + 63)).dx`
= `int (x - 7)/sqrt(x^2 - 16x + 63).dx`
Let x – 7 = `"A"[d/dx(x^2 - 16x + 63)] + "B"`
= A(2x – 16) + B
= 2Ax + (B – 16A)
Comparing the coefficient of x and constant term on both sides, we get
2A = 1
∴ A = `(1)/(2)` and
B – 16A = – 7
∴ `"B" - 16(1/2)` = – 7
∴ B = 1
∴ x – 7 = `(1)/(2)(2x - 16) + 1`
∴ I = `int (1/2(2x - 16) + 1)/sqrt(x^2 - 16x + 63).dx`
= `(1)/(2) int (2x - 16)/sqrt(x^2 - 16x + 63).dx + int (1)/sqrt(x^2 - 16x + 63).dx`
= `(1)/(2)"I"_1 + "I"_2`
In I1, put x2 – 16x + 63 = t
∴ (2x – 16)dx = dt
∴ I1 = `(1)/(2) int (1)/sqrt(t)dt`
= `(1)/(2) int t^(-1/2)dt`
= `(1)/(2) t^(1/2)/((1/2)) + c_1`
= `sqrt(x^2 - 16x + 63) + c_1`
I2 = `int (1)/sqrt(x^2 - 16x + 63).dx`
= `int (1)/sqrt((x - 8)^2 - 1^2).dx`
= `log|x - 8 + sqrt((x - 8)^2 - 1^2)| + c_2`
= `log|x - 8 + sqrt(x^2 - 16x + 63)| + c_2`
∴ I = `sqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`,, where c = c1 + c2.
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 `intsqrtx/sqrt(a^3-x^3)dx`
Integrate the functions:
`(2x)/(1 + x^2)`
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
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 sin 4x cos 3x dx`
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 : `(1)/(sqrt(x) + sqrt(x^3)`
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Integrate the following functions w.r.t. x : tan5x
Integrate the following functions w.r.t. x : sin5x.cos8x
Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
`int sqrt(1 + "x"^2) "dx"` =
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 "x - 1"/sqrt("x + 4")` dx
Evaluate: `int 1/(sqrt("x") + "x")` dx
`int sqrt(1 + sin2x) dx`
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
Evaluated the following
`int x^3/ sqrt (1 + x^4 )dx`
Evaluate `int(1+ x + x^2/(2!)) dx`
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate:
`int(sqrt(tanx) + sqrt(cotx))dx`
`int "cosec"^4x dx` = ______.
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate `int(5x^2-6x+3)/(2x-3) dx`
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
