Mathematics : asked on rosie20052019

21.04.2022

Integral of x/(x + 1)²(x+2)²

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09.07.2023, solved by verified expert

Rajat Thapa

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Mathematics, DAV Post Graduate College

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I suppose you mean the integral

Break up the integrand into partial fractions:

Solve for the coefficients:

So we have

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Mathematics

Integral of x/(x + 1)²(x+2)²

Step-by-step answer

P Answered by Specialist

I suppose you mean the integral

$\displaystyle \int \frac{x}{(x+1)^2 (x+2)^2} \, dx$

Break up the integrand into partial fractions:

$\dfrac{x}{(x+1)^2 (x+2)^2} = \dfrac{a}{x+1} + \dfrac{b}{(x+1)^2} + \dfrac{c}{x+2} + \dfrac{d}{(x+2)^2}$

$\dfrac{x}{(x+1)^2 (x+2)^2} \\ = \dfrac{a(x+1)(x+2)^2 + b(x+2)^2 + c(x+1)^2(x+2) + d(x+1)^2}{(x+1)^2 (x+2)^2}$

$\dfrac{x}{(x+1)^2 (x+2)^2} \\ = \dfrac{(a+c)x^3+(5a+b+4c+d)x^2+(8a+4b+5c+2d)x + 4a+4b+2c+d}{(x+1)^2(x+2)^2}$

x = (a+c)x^3+(5a+b+4c+d)x^2+(8a+4b+5c+2d)x + 4a+4b+2c+d

Solve for the coefficients:

$\begin{cases}a+c=0 \\ 5a+b+4c+d=0 \\ 8a+4b+5c+2d=1 \\ 4a+4b+2c+d=0\end{cases} \implies a=3,b=-1,c=-3,d=-2$

So we have

$\displaystyle \int \frac{x}{(x+1)^2 (x+2)^2} \, dx= \int \left(\frac3{x+1} - \frac1{(x+1)^2} - \frac3{x+2} - \frac2{(x+2)^2}\right) \, dx$

$\displaystyle \int \frac{x}{(x+1)^2 (x+2)^2} \, dx= 3\ln|x+1| + \frac1{x+1} - 3\ln|x+2| + \frac2{x+2} + C$

$\displaystyle \int \frac{x}{(x+1)^2 (x+2)^2} \, dx= \ln|x+1|^3 - \ln|x+2|^3 + \frac{3x+4}{(x+1)(x+2)} + C$

$\displaystyle \int \frac{x}{(x+1)^2 (x+2)^2} \, dx= \boxed{\ln\left|\frac{x+1}{x+2}\right|^3 + \frac{3x+4}{(x+1)(x+2)} + C}$

Mathematics

With cal 2 exercise involving integrals? i have to find the arc length of y = lnx given that 1 ≤ x ≤ sqrt (3) here is my procedure: i found the derivative of y = lnx and got, y = 1 / (x) 1 + (1/(x))^2 = 1 + (1/(x^2)) = (x^2+1)/(x^2) integral from 1 to sqrt 3 of sqrt((x^2+1/(x^2)) dx = sqrt ( x ^ 2 + 1 ) / ( x ) dx x = tan b dx = (sec^2)bdb sqrt(x^2+1) = sqrt ( (tan^2)b + 1) = sqrt ( (sec^2)b) = secb integral = ((sec)/(tanb)) times (sec^2)bdb and here comes the part i don t understand, in my exercise it becomes integral = ((sec^2)b)/((tan^2)b))

Step-by-step answer

P Answered by PhD

$\displaystyle\int\frac{\sqrt{x^2+1}}x\,\mathrm dx=\int\frac{\sqrt{\tan^2B+1}}{\tan B}\sec^2B\,\mathrm dB$

$\tan^2B+1=\sec^2B$

$\sqrt{\sec^2B}=\sec B$ (provided that $\sec B0$ )

Then

$\dfrac{\sec B}{\tan B}\cdot\sec^2B=\dfrac{\sec^2B}{\tan B}\cdot\sec B$

(just moving around a factor of $\sec B$ )

$\dfrac{\sec B}{\tan B}\cdot\sec^2B=\dfrac{\sec^2B}{\tan^2B}\cdot\sec B\tan B$

(multiply by $\dfrac{\tan B}{\tan B}$ )

Mathematics

Determine whether each integral is convergent or divergent. 1- 1/(x-2)^3/2 dx, limits ( infinite to 3) 2- (1/3-4x)dx, limits (0 to -infinite) 3- e^(-5p) dx, limits ( infinite to 2) 4- (x^2/(sqrt(1+x^3)))dx , limits ( infinite to 0) 5- lnx/x dx , limits(infinite to 1) 6- 1/(x^2 +x)dx , limits (infinite to 1) 7- 3/x^5 dx ,limits (1 to 0) 8- dx/(x+2)^1/4 , limits (14 to -2) 9- 1/(x-1)^1/3 , limits (9 to 0) 10- e^x /((e^x) -1), limits (1 to -1) 11- z^2 lnz dz ,limits( 2 to 0) 12- s={(x,y) | x = 1,0 = y =e^-x}

Step-by-step answer

P Answered by PhD

Divergers

Step-by-step explanation:

Mathematics

Step-by-step answer

P Answered by PhD

$\displaystyle\int\frac{\sqrt{x^2+1}}x\,\mathrm dx=\int\frac{\sqrt{\tan^2B+1}}{\tan B}\sec^2B\,\mathrm dB$

$\tan^2B+1=\sec^2B$

$\sqrt{\sec^2B}=\sec B$ (provided that $\sec B0$ )

Then

$\dfrac{\sec B}{\tan B}\cdot\sec^2B=\dfrac{\sec^2B}{\tan B}\cdot\sec B$

(just moving around a factor of $\sec B$ )

$\dfrac{\sec B}{\tan B}\cdot\sec^2B=\dfrac{\sec^2B}{\tan^2B}\cdot\sec B\tan B$

(multiply by $\dfrac{\tan B}{\tan B}$ )

Mathematics

Consider the following initial-value problem. (x + y)2 dx + (2xy + x2 − 2) dy = 0, y(1) = 1 Let ∂f ∂x = (x + y)2 = x2 + 2xy + y2. Integrate each term of this partial derivative with respect to x, letting h(y) be an

Step-by-step answer

P Answered by PhD

$(x+y)^2\,\mathrm dx+(2xy+x^2-2)\,\mathrm dy=0$

Suppose the ODE has a solution of the form F(x,y)=C , with total differential

$\dfrac{\partial F}{\partial x}\,\mathrm dx+\dfrac{\partial F}{\partial y}\,\mathrm dy=0$

This ODE is exact if the mixed partial derivatives are equal, i.e.

$\dfrac{\partial^2F}{\partial y\partial x}=\dfrac{\partial^2F}{\partial x\partial y}$

We have

$\dfrac{\partial F}{\partial x}=(x+y)^2\implies\dfrac{\partial^2F}{\partial y\partial x}=2(x+y)$

$\dfrac{\partial F}{\partial y}=2xy+x^2-2\implies\dfrac{\partial^2F}{\partial x\partial y}=2y+2x=2(x+y)$

so the ODE is indeed exact.

Integrating both sides of

$\dfrac{\partial F}{\partial x}=(x+y)^2$

with respect to gives

$F(x,y)=\dfrac{(x+y)^3}3+g(y)$

Differentiating both sides with respect to gives

$\dfrac{\partial F}{\partial y}=2xy+x^2-2=(x+y)^2+\dfrac{\mathrm dg}{\mathrm dy}$

$\implies x^2+2xy-2=x^2+2xy+y^2+\dfrac{\mathrm dg}{\mathrm dy}$

$\implies\dfrac{\mathrm dg}{\mathrm dy}=-y^2-2$

$\implies g(y)=-\dfrac{y^3}3-2y+C$

$\implies F(x,y)=\dfrac{(x+y)^3}3-\dfrac{y^3}3-2y+C$

so the general solution to the ODE is

$F(x,y)=\dfrac{(x+y)^3}3-\dfrac{y^3}3-2y=C$

Given that y(1)=1 , we find

$\dfrac{(1+1)^3}3-\dfrac{1^3}3-2=C\implies C=\dfrac13$

so that the solution to the IVP is

$F(x,y)=\dfrac{(x+y)^3}3-\dfrac{y^3}3-2y=\dfrac13$

$\implies\boxed{(x+y)^3-y^3-6y=1}$

Mathematics

Step-by-step answer

P Answered by PhD

$(x+y)^2\,\mathrm dx+(2xy+x^2-2)\,\mathrm dy=0$

Suppose the ODE has a solution of the form F(x,y)=C , with total differential

$\dfrac{\partial F}{\partial x}\,\mathrm dx+\dfrac{\partial F}{\partial y}\,\mathrm dy=0$

This ODE is exact if the mixed partial derivatives are equal, i.e.

$\dfrac{\partial^2F}{\partial y\partial x}=\dfrac{\partial^2F}{\partial x\partial y}$

We have

$\dfrac{\partial F}{\partial x}=(x+y)^2\implies\dfrac{\partial^2F}{\partial y\partial x}=2(x+y)$

$\dfrac{\partial F}{\partial y}=2xy+x^2-2\implies\dfrac{\partial^2F}{\partial x\partial y}=2y+2x=2(x+y)$

so the ODE is indeed exact.

Integrating both sides of

$\dfrac{\partial F}{\partial x}=(x+y)^2$

with respect to gives

$F(x,y)=\dfrac{(x+y)^3}3+g(y)$

Differentiating both sides with respect to gives

$\dfrac{\partial F}{\partial y}=2xy+x^2-2=(x+y)^2+\dfrac{\mathrm dg}{\mathrm dy}$

$\implies x^2+2xy-2=x^2+2xy+y^2+\dfrac{\mathrm dg}{\mathrm dy}$

$\implies\dfrac{\mathrm dg}{\mathrm dy}=-y^2-2$

$\implies g(y)=-\dfrac{y^3}3-2y+C$

$\implies F(x,y)=\dfrac{(x+y)^3}3-\dfrac{y^3}3-2y+C$

so the general solution to the ODE is

$F(x,y)=\dfrac{(x+y)^3}3-\dfrac{y^3}3-2y=C$

Given that y(1)=1 , we find

$\dfrac{(1+1)^3}3-\dfrac{1^3}3-2=C\implies C=\dfrac13$

so that the solution to the IVP is

$F(x,y)=\dfrac{(x+y)^3}3-\dfrac{y^3}3-2y=\dfrac13$

$\implies\boxed{(x+y)^3-y^3-6y=1}$

Mathematics

If f(x)=integral of sqrt(t^3+1) dt from 0 to x, then f (2)=?

Step-by-step answer

P Answered by Specialist

F'(2) = 3

$\int\limits^x_0 {\sqrt{t^3+1}} \, dt$

Using Fundamental Theorem of Calculus:

Since x = 2, you now have:

$\int\limits^2_0 {\sqrt{t^3+1} } \, dt$

so F'(x) = $\sqrt{t^3+1}$

--> F'(2) = $\sqrt{2^3+1}$ = $\sqrt{9}$ = 3

so F'(2) = 3

Mathematics

If f(x)=integral of sqrt(t^3+1) dt from 0 to x, then f (2)=?

Step-by-step answer

P Answered by Specialist

F'(2) = 3

$\int\limits^x_0 {\sqrt{t^3+1}} \, dt$

Using Fundamental Theorem of Calculus:

Since x = 2, you now have:

$\int\limits^2_0 {\sqrt{t^3+1} } \, dt$

so F'(x) = $\sqrt{t^3+1}$

--> F'(2) = $\sqrt{2^3+1}$ = $\sqrt{9}$ = 3

so F'(2) = 3

Mathematics

Example 3 (a) set up the integral for the length of the arc of the hyperbola xy = 3 from the point (1, 3) to the point 6, 1 2 . (b) use simpson s rule with n = 10 to estimate the arc length. solution (a) we have y = 3 x dy dx = and so the arc length is l = 6 1 + dy dx 2 dx 1 = 6 1 + 9 x4 dx 1 = 6 x4 + 9 x2 dx 1 . (b) using simpson s rule with a = 1, b = 6, n = 10, δx = 0.5, and f(x) = , we have l = 6 1 + 9 x4 dx 1 ≈ δx 3 [f(1) + 4f(1.5) + 2f(2) + 4f(2.5) + ⋯ + 2f(5) + 4f(5.5) + f(6)] ≈ (rounded to four decimal places).

Step-by-step answer

P Answered by Master

Step-by-step explanation:

Given is the function

xy =3

Use product rule to find derivative

xy'+y =0 OR $y'=\frac{-y}{x}$

Arc length = $\int\limits^1_6 {\sqrt{1+\frac{y^2}{x^2} } } \, dx \\=\int\limits^1_6 {\sqrt{\frac{x^2+y^2}{x^2} } } \, dx \\\\$

Substitute y = 3/x

WE get

arc length = $\int\limits^1_6 {\sqrt{1+\frac{9}{x^4} } } \, dx \\\\=\int\limits^1_6 {\sqrt{\frac{x^4+9}{x^4} } } \, dx \\$

Using simpson rule for 1/3 we have

x(x^4+9)^1/2f(x)2 times or 4 times

13.162277663.162277663.16227766

1.53.751.6666666676.666666667

251.25

2.56.9327123121.109233974.43693588

39.4868329811.054092553

3.512.611998261.0295508784.118203512

416.27882061.0174262872.034852575

4.520.47101611.0109143754.0436575

525.179356621.0071742652.01434853

5.530.398396341.0049056644.019622656

636.124783741.0034662152.00693243

6.542.356374961.0025177514.010071002

749.091750831.0018724662.003744932

7.556.329943191.0014212124.005684849

864.070273921.001098032.00219606

8.572.312256911.0008616874.00344675

981.055536521.0006856362.001371272

9.590.299847731.0005523294.002209318

10100.04498991.0004498992.000899798

10.5110.29080881.0003701484.00148059

11121.03718441.0003073092.000614618

11.5132.28402211.0002572564.001029023

12144.03124661.000216994.82148E-05

52.270413638.711735605

Answer is 8.7117

Mathematics

Let f(x,y)= (y^2+1i + (2xy-2)j. compute single integral with subscript of c f * dr where a). c is the straight path from (0,0) to (0,1) b,) c is the path from (0,0) to (0,1) along the parabola y=x^2

Step-by-step answer

P Answered by Specialist

I do not know if this is exactly what you are looking for

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