Advertisements
Advertisements
प्रश्न
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Advertisements
उत्तर
Let I = `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Let `1/((x^2 + 1)(x^2 + 2)) = A/(x^2 + 1) + B/(x^2 + 2)`
⇒ 1 = A(x2 + 2) + B(x2 + 1)
⇒ 1 = (A + B)x2 + (2A + B)
On comparing both sides, we get
A + B = 0 and 2A + B = 0
On solving the above equations, we get
A = 1 and B = –1
∴ I = `int(1/(x^2 + 1) - 1/(x^2 + 2))2xdx`
I = `int (2x)/(x^2 + 1) dx - int (2x)/(x^2 + 2) dx`
I = `log|x^2 + 1| - log|x^2 + 2| + C`
I = `log|(x^2 + 1)/(x^2 + 2)| + C`
APPEARS IN
संबंधित प्रश्न
Integrate the function in x sin x.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Find :
`∫(log x)^2 dx`
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int(x + 1/x)^3 dx` = ______.
Evaluate `int 1/(x(x - 1)) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int log x * [log ("e"x)]^-2` dx = ?
`int tan^-1 sqrt(x) "d"x` is equal to ______.
If u and v ore differentiable functions of x. then prove that:
`int uv dx = u intv dx - int [(du)/(d) intv dx]dx`
Hence evaluate `intlog x dx`
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
`int1/sqrt(x^2 - a^2) dx` = ______
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
`inte^(xloga).e^x dx` is ______
Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate `int(1 + x + x^2/(2!))dx`.
