Advertisements
Advertisements
प्रश्न
Prove the following:
`int_1^3 dx/(x^2(x +1)) = 2/3 + log 2/3`
Advertisements
उत्तर
Let `I = int_1^3 dx/(x^2 (x + 1))`
Now, `1/(x^2 (x + 1)) = A/x + B/x^2 + C/(x + 1)`
∴ 1 ≡ Ax (x + 1) + B(x + 1) + Cx2 ....(i)
Putting x = 0 in (i), we get
1 = B (0 + 1)
⇒ B = 1
Putting x = -1 in (i), we get
= C (-1)2
⇒ C = 1
Comparing coefficients of x2 on the sides of (i), we get
∴ 0 = A + C
∴ A = - C = - 1
⇒ A = -1
∴ `1/(x^2 (x + 1)) = (- 1)/x + 1/x^2 + 1/(x + 1)`
∴ `int_1^3 1/(x^2 (x + 1))`dx
`= - int_1^3 1/x "dx" + int_1^3 1/x^2 "dx" + int_1^3 1/(x + 1)`dx
`= [- log |x| + x^-1/-1 + log |x + 1|]_1^3`
`= [- 1/x + log |(x + 1)/x|]_1^3 = (-1/3 + 1) + log 4/3 - log 2`
`= 2/3 + log (4/3 xx 1/2)`
`= 2/3 + log 2/3`
APPEARS IN
संबंधित प्रश्न
Evaluate `int_(-1)^2(e^3x+7x-5)dx` as a limit of sums
Evaluate `int_1^3(e^(2-3x)+x^2+1)dx` as a limit of sum.
Evaluate the following definite integrals as limit of sums.
`int_0^5 (x+1) dx`
Evaluate the following definite integrals as limit of sums.
`int_0^4 (x + e^(2x)) dx`
Evaluate the definite integral:
`int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx`
Evaluate the definite integral:
`int_0^(pi/2) (cos^2 x dx)/(cos^2 x + 4 sin^2 x)`
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
Prove the following:
`int_0^(pi/2) sin^3 xdx = 2/3`
Prove the following:
`int_0^1sin^(-1) xdx = pi/2 - 1`
`int dx/(e^x + e^(-x))` is equal to ______.
If f (a + b - x) = f (x), then `int_a^b x f(x )dx` is equal to ______.
\[\int\frac{1}{x} \left( \log x \right)^2 dx\]
\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]
Evaluate the following integral:
\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]
Evaluate the following as limit of sum:
`int _0^2 (x^2 + 3) "d"x`
Evaluate the following as limit of sum:
`int_0^2 "e"^x "d"x`
Evaluate the following:
`int_0^1 (x"d"x)/sqrt(1 + x^2)`
Evaluate the following:
`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))` (Hint: Let x = sin θ)
Evaluate the following:
`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2) "d"x`
The value of `int_(-pi)^pi sin^3x cos^2x "d"x` is ______.
The value of `lim_(x -> 0) [(d/(dx) int_0^(x^2) sec^2 xdx),(d/(dx) (x sin x))]` is equal to
What is the derivative of `f(x) = |x|` at `x` = 0?
The value of `lim_(n→∞)1/n sum_(r = 0)^(2n-1) n^2/(n^2 + 4r^2)` is ______.
`lim_(n→∞){(1 + 1/n^2)^(2/n^2)(1 + 2^2/n^2)^(4/n^2)(1 + 3^2/n^2)^(6/n^2) ...(1 + n^2/n^2)^((2n)/n^2)}` is equal to ______.
