Advertisements
Advertisements
प्रश्न
Integrate the function:
`(5x)/((x+1)(x^2 +9))`
Advertisements
उत्तर
Let I = `int (5x)/((x + 1)(x^2 + 9))`dx
`therefore (5x)/((x + 1)(x^2 + 9)) = A/(x + 1) + (Bx + C)/(x^2 + 9)`
`=> 5x = A(x^2 + 9) + (Bx + C)(x + 1)`
Putting x = -1 in equation (1),
- 5 = A(1 + 9)
⇒ - 5 = 10 A
`therefore A = - 5/10 = - 1/2`
From equation (1),
Comparing the coefficients of x2 and the constant term,
0 = A + B
⇒ B = - A = `1/2`
0 = 9A + C
⇒ C = - 9A = `9/2`
`therefore (5x)/((x + 1)(x^2 + 9)) = (- 1)/(2(x + 1)) + (1/2 x + 9/2)/(x^2 + 9)`
∴ `I = int(1/2)/(x + 1) dx + int (1/2 x + 9/2)/(x^2 + 9) dx`
`= -1/2 log (x + 1) + 1/4 int (2x)/(x^2 + 9) dx + 9/2 int dx/ (x^2 + 3^2) + C`
`= -1/2 log (x + 1) + 1/4 log (x^2 + 9) + 9/2 xx 1/3 tan^-1 x/3 + C`
`= -1/2 log (x + 1) + 1/4 log (x^2 + 9) + 3/2 tan^-1 x/3 + C`
APPEARS IN
संबंधित प्रश्न
Write the antiderivative of `(3sqrtx+1/sqrtx).`
If `f(x) =∫_0^xt sin t dt` , then write the value of f ' (x).
Find an anti derivative (or integral) of the following function by the method of inspection.
Cos 3x
Find the following integrals:
`int (4e^(3x) + 1)`
Find the following integrals:
`int (x^3 + 5x^2 -4)/x^2 dx`
Find the following integrals:
`int (x^3 + 3x + 4)/sqrtx dx`
Find the following integrals:
`int (x^3 - x^2 + x - 1)/(x - 1) dx`
Find the following integrals:
`int(2x^2 - 3sinx + 5sqrtx) dx`
Find the following integrals:
`intsec x (sec x + tan x) dx`
Find the following integrals:
`int (2 - 3 sinx)/(cos^2 x) dx.`
The anti derivative of `(sqrtx + 1/ sqrtx)` equals:
If `d/dx f(x) = 4x^3 - 3/x^4` such that f(2) = 0, then f(x) is ______.
Integrate the function:
`1/(xsqrt(ax - x^2)) ["Hint : Put x" = a/t]`
Integrate the function:
`1/(x^2(x^4 + 1)^(3/4))`
Integrate the function:
`sinx/(sin (x - a))`
Integrate the function:
`cos x/sqrt(4 - sin^2 x)`
Integrate the function:
`(sin^8 x - cos^8 x)/(1-2sin^2 x cos^2 x)`
Integrate the function:
`1/(cos (x+a) cos(x+b))`
Integrate the function:
`cos^3 xe^(log sinx)`
Integrate the function:
`tan^(-1) sqrt((1-x)/(1+x))`
Integrate the function:
`(sqrt(x^2 +1) [log(x^2 + 1) - 2log x])/x^4`
Find : \[\int\frac{\left( x^2 + 1 \right)\left( x^2 + 4 \right)}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)}dx\] .
Evaluate: `int (1 - cos x)/(cos x(1 + cos x)) dx`
If `d/(dx) f(x) = 4x^3 - 3/x^4`, such that `f(2) = 0`, then `f(x)` is
`int (dx)/sqrt(9x - 4x^2)` equals
`int (xdx)/((x - 1)(x - 2))` equals
`int sqrt(1 + x^2) dx` is equal to
What is anti derivative of `e^(2x)`
If the normal to the curve y(x) = `int_0^x(2t^2 - 15t + 10)dt` at a point (a, b) is parallel to the line x + 3y = –5, a > 1, then the value of |a + 6b| is equal to ______.
`d/(dx)x^(logx)` = ______.
If y = `x^((sinx)^(x^((sinx)^(x^(...∞)`, then `(dy)/(dx)` at x = `π/2` is equal to ______.
`int (dx)/sqrt(5x - 6 - x^2)` equals ______.
Anti-derivative of `(tanx - 1)/(tanx + 1)` with respect to x is ______.
