Advertisements
Advertisements
प्रश्न
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
Advertisements
उत्तर
Let I = `int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
Let 3x + 4 = `"A" "d"/("d"x)(2x^2 + 2x + 1) + "B"`
∴ 3x + 4 = A(4x + 2) + B
∴ 3x + 4 = 4Ax + 2A + B
By equating the coefficients on both sides, we get
4A = 3 and 2A + B = 4
∴ A = `3/4` and `2(3/4) + "B"` = 4
∴ B = `5/2`
∴ 3x + 4 = `3/4(4x + 2) + 5/2`
∴ I = `int (3/4(4x + 2) + 5/2)/sqrt(2x^2 + 2x + 1) "d"x`
= `3/4 int (4x + 2)/sqrt(2x^2 + 2x + 1) "d"x + 5/2 int 1/sqrt(2x^2 + 2x + 1) "d"x`
= I1 + I2 ........(i)
I1 = `3/4 int (4x + 2)/sqrt(2x^2 + 2x + 1) "d"x`
Put 2x2 + 2x + 1 = t
∴ (4x + 2) dx = dt
∴ I1 = `3/4 int "dt"/sqrt("t")`
= `3/4 int "t"^(1/2) "dt"`
= `3/4("t"^(1/2)/(1/2)) + "c"_1`
= `3/2 sqrt("t") + "c"_1`
∴ I1 = `3/2 sqrt(2x^2 + 2x + 1) + "c"_1` .........(ii)
I2 = `5/2 int 1/sqrt(2x^2 + 2x + 1) "d"x`
= `5/2 int 1/sqrt(2(x^2 + x + 1/2)) "d"x`
`(1/2 "coefficient of" x)^2 = (1/2 xx 1)^2`
= `1/4`
∴ I2 = `5/(2sqrt(2)) int 1/sqrt(x^2 + x + 1/4 - 1/4 + 1/2) "d"x`
= `5/(2sqrt(2)) int 1/sqrt((x + 1/2)^2 - (1/2)^2) "d"x`
= `5/(2sqrt(2)) log|x + 1/2 + sqrt((x + 1/2)^2 - (1/2)^2)| + "c"_2`
∴ I2 = `5/(2sqrt(2)) log|x + 1/2 + sqrt(x^2 + x + 1/2)| + "c"_2` ........(iii)
From (i), (ii) and (iii), we get
I = `3/2 sqrt(2x^2 + 2x + 1) + 5/(2sqrt(2)) log|x + 1/2 + sqrt(x^2 + x + 1/2)| + "c"`,
where c = c1 + c2
APPEARS IN
संबंधित प्रश्न
Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`
Find : `int x^2/(x^4+x^2-2) dx`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
`int (xdx)/((x - 1)(x - 2))` equals:
Find :
`∫ sin(x-a)/sin(x+a)dx`
Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int (sinx)/(sin3x) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int x^3tan^(-1)x "d"x`
`int 1/(sinx(3 + 2cosx)) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
State whether the following statement is True or False:
For `int (x - 1)/(x + 1)^3 "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
`int x/((x - 1)^2 (x + 2)) "d"x`
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate: `int (dx)/(2 + cos x - sin x)`
If f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
If \[\int\frac{2x+3}{(x-1)(x^{2}+1)}\mathrm{d}x\] = \[=\log_{e}\left\{(x-1)^{\frac{5}{2}}\left(x^{2}+1\right)^{a}\right\}-\frac{1}{2}\tan^{-1}x+\mathrm{A}\] where A is an arbitrary constant, then the value of a is
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
