Advertisements
Advertisements
प्रश्न
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
Advertisements
उत्तर
Let `I = int(5x^2 + 4x +7)/(2x + 3)^(3/2) dx`
Put 2x + 3 = t2 ...(i)
Differentiating w.r.t. x, we get
2dx = 2t dt
∴ dx = t dt
From (i), we get
`x = (t^2 - 3)/2`
∴ `I = int (5((t^2 - 3)/2)^2 + 4((t^2 - 3)/2) + 7)/(t^2)^(3/2) * t dt`
`I = int (5((t^4 - 6t^2 + 9)/4) + 2t^2 - 6 + 7)/t^3 * t dt`
`I = int (5t^4 - 30t^2 + 45 + 8t^2 + 4)/(4t^3) * t dt`
`I = int (5t^4 - 22t^2 + 49)/(4t^2) dt`
`I = 5/4 int t^2 dt - 22/4 int dt + 49/4 int t^(-2) dt`
`I = 5/4 * t^3/3 - 22/4 t + 49/4 * t^(-1)/(-1) + c`
`I = 5/12t^3 - 11/2t - 49/(4t) + c`
∴ `I = 5/12(2x + 3)^(3/2) - 11/2 sqrt(2x + 3) - 49/4 * 1/sqrt(2x + 3) + c`
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
`int (xdx)/((x - 1)(x - 2))` equals:
`int (dx)/(x(x^2 + 1))` equals:
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) 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 : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`
Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
State whether the following statement is True or False.
If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.
`int x^7/(1 + x^4)^2 "d"x`
`int 1/(x(x^3 - 1)) "d"x`
`int (sinx)/(sin3x) "d"x`
`int sec^3x "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. Find the fraction.
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
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(2x^3 - 1)/(x^4 + x)dx`
