Advertisements
Advertisements
प्रश्न
Find :`int(x^2+x+1)/((x^2+1)(x+2))dx`
Advertisements
उत्तर
Consider the given function
`I=int(x^2+x+1)/((x^2+1)(x+2))dx`
Let `(x^2+x+1)/((x^2+1)(x+2))=A/(x+2)+(Bx+C)/(x^2+1)`
`= (A(x^2+1)+(Bx+C)(x+2))/((x^2+1)(x+2))`
`=((A+B)x^2+(2B+C)x+(2C+A))/((x^2+1)(x+2))`
Thus equating the coefficients, we have,
A+B=1...(1)
2B+C=1...(2)
2C+A=1...(3)
Solving the above three equations, we have,
`A=3/2,B=2/5 `
`:.(x^2+x+1)/((x^2+1)(x+2))=A/(x+2)+(Bx+C)/(x^2+1)`
`=>(x^2+x+1)/((x^2+1)(x+2))=3/(5(x+2))+(2x+1)/(5(x^2+1)`
`:.I=int(x^2+x+1)/((x^2+1)(x+2))dx`
`=int[3/(5(x+2))+(2x+1)/(5(x^2+1))]dx`
`=3/5intdx/((x+2))dx+1/5int(2x+1)/((x^2+1))dx`
`=3/5log(x+2)+1/5int(2x)/((x^2+1))dx+1/5intdx/((x^2+1))`
`=3/2log(x+2)+1/5log(x^2+1)+1/5tan^(-1)x+C`
APPEARS IN
संबंधित प्रश्न
Write the antiderivative of `(3sqrtx+1/sqrtx).`
Find an anti derivative (or integral) of the following function by the method of inspection.
Cos 3x
Find an anti derivative (or integral) of the following function by the method of inspection.
e2x
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`
Integrate the function:
`1/(xsqrt(ax - x^2)) ["Hint : Put x" = a/t]`
Integrate the function:
`(5x)/((x+1)(x^2 +9))`
Integrate the function:
`cos x/sqrt(4 - sin^2 x)`
Integrate the function:
`1/((x^2 + 1)(x^2 + 4))`
Integrate the function:
`e^(3log x) (x^4 + 1)^(-1)`
Integrate the function:
`1/sqrt(sin^3 x sin(x + alpha))`
Integrate the function:
`sqrt((1-sqrtx)/(1+sqrtx))`
If `d/(dx) f(x) = 4x^3 - 3/x^4`, such that `f(2) = 0`, then `f(x)` is
`sqrt((10x^9 + 10^x log e^10)/(x^10 + 10^x)) dx` equals
`int (dx)/(sin^2x cos^2x) dx` equals
`int (e^x (1 + x))/(cos^2 (xe^x)) dx` equal
`int (dx)/(x(x^2 + 1))` equals
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 ______.
`int (dx)/sqrt(5x - 6 - x^2)` equals ______.
What is integration called because it reverses the operation of finding derivatives?
Which statement about indefinite integrals is correct?
Evaluate \[\int \sin x\,dx\]
Evaluate \[\int \sec^2 x\,dx\]
Evaluate \[\int \sec x\tan x\,dx\]
Evaluate \[\int \text{cosec} x\cot x\,dx\]
Which expression is an antiderivative of \[\frac{1}{\sqrt{1-x^2}}\]?
Evaluate \[\int e^x\,dx\]
Evaluate \[\int a^x\,dx\]
