Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Advertisements
उत्तर
Let I = `int (3x + 1)/sqrt(-2x^2 + x + 3).dx`
Let 3x + 1 = `"A"[d/dx(-2x^2 + x + 3)] + "B"`
= A(2 – 2x) + B
∴ 3x + 1 = 2Ax + (2A + B)
Comparing the coefficient of x and constant on both the sides, we get
– 2A = 7 and 2A + B = 3
∴ A = `(-7)/(2) and 2(-7/2) + "B" ` = 3
∴ B = 10
∴ 7x + 3 = `(-7)/(2)(2 - 2x) + 10`
∴ I = `int ((-7)/(2)(2 - 2x) + 10)/sqrt(3 + 2x - x^2).dx`
= `(-7)/(2) int ((2 - 2x))/sqrt(3 + 2x - x^2).dx + 10 int(1)/sqrt(3 + 2x - x^2)x`
= `(-7)/(2)"I"_1 + 10"I"_2`
In I1, put 3 + 2x – x2 = t
∴ (2 – 2x)dx = dt
∴ I1 = `int (1)/sqrt(t)dt`
= `int t^(-1/2) dt`
= `t^(1/2)/(1/2) + c_1`
= `2sqrt(3 + 2x - x^2) + c_1`
I2 = `int (1)/sqrt(3 - (x^2 - 2x + 1) + 1).dx`
= `int (1)/sqrt((2)^2 - (x - 1)^2).dx`
= `sin^-1((x - 1)/2) + c_2`
`-(3)/(2) sqrt(-2x^2 + x + 3) + (7)/(4sqrt(2)) sin^-1((4x - 1)/5) + c`.
APPEARS IN
संबंधित प्रश्न
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`cos x /(sqrt(1+sinx))`
Integrate the functions:
cot x log sin x
Write a value of
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Evaluate the following integrals : tan2x dx
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`
Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x : tan5x
Evaluate the following : `int (1)/(4x^2 - 3).dx`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Choose the correct options from the given alternatives :
`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` 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 = ?
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"x" log"x")` dx
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
`int 1/(cos x - sin x)` dx = _______________
`int (sin4x)/(cos 2x) "d"x`
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
`int_1^3 ("d"x)/(x(1 + logx)^2)` = ______.
If f'(x) = `x + 1/x`, then f(x) is ______.
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate:
`int sqrt((a - x)/x) dx`
`int "cosec"^4x dx` = ______.
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
