Math Problem Statement

Evaluate the integral ∫ from 1 to e of coth^2(ln(x) + x)(x + 1)/x dx.

Solution

To solve the integral

1ecoth2(lnx+x)(x+1)xdx,\int_1^e \frac{\text{coth}^2(\ln x + x)(x + 1)}{x} \, dx,

let's proceed step by step:


Step 1: Simplify the integral

The hyperbolic cotangent squared function is defined as: coth2u=cosh2usinh2u.\text{coth}^2 u = \frac{\cosh^2 u}{\sinh^2 u}.

The integral becomes: 1ecosh2(lnx+x)sinh2(lnx+x)(x+1)xdx.\int_1^e \frac{\cosh^2(\ln x + x)}{\sinh^2(\ln x + x)} \cdot \frac{(x+1)}{x} \, dx.


Step 2: Change of variables

Let u=lnx+xu = \ln x + x. Then, the derivative is: du=(1x+1)dx=x+1xdx.du = \left(\frac{1}{x} + 1\right) dx = \frac{x + 1}{x} dx.

Substituting this, the integral becomes: coth2udu.\int \text{coth}^2 u \, du.


Step 3: Integrate coth2u\text{coth}^2 u

Recall that: coth2u=1+\csch2u,\text{coth}^2 u = 1 + \csch^2 u, where \csch2u\csch^2 u is the hyperbolic cosecant squared. Thus: coth2udu=1du+\csch2udu.\int \text{coth}^2 u \, du = \int 1 \, du + \int \csch^2 u \, du.

From hyperbolic integrals, we know: 1du=uand\csch2udu=cothu.\int 1 \, du = u \quad \text{and} \quad \int \csch^2 u \, du = -\coth u.

Hence: coth2udu=ucothu+C.\int \text{coth}^2 u \, du = u - \coth u + C.


Step 4: Back-substitute

Returning to u=lnx+xu = \ln x + x, we have: 1ecoth2(lnx+x)(x+1)xdx=[(lnx+x)coth(lnx+x)]1e.\int_1^e \frac{\text{coth}^2(\ln x + x)(x+1)}{x} \, dx = \left[ (\ln x + x) - \coth(\ln x + x) \right]_1^e.


Step 5: Evaluate the bounds

At x=ex = e: u=lne+e=1+e.u = \ln e + e = 1 + e.

At x=1x = 1: u=ln1+1=1.u = \ln 1 + 1 = 1.

Thus: [ \int_1^e \frac{\text{coth}^2(\ln x + x)(x+1)}{x} , dx = \left[(1+e) - \coth(1+e)\right] - \left[1 - \coth(1)\right]. ]

Simplify: =(1+ecoth(1+e))(1coth(1)).= (1+e - \coth(1+e)) - (1 - \coth(1)).

Final result: 1ecoth2(lnx+x)(x+1)xdx=ecoth(1+e)+coth(1).\int_1^e \frac{\text{coth}^2(\ln x + x)(x+1)}{x} \, dx = e - \coth(1+e) + \coth(1).


Would you like further clarification or a numerical evaluation of this result?

Related Questions:

  1. What are the steps for deriving the hyperbolic trigonometric identities used here?
  2. How do we handle integrals involving hyperbolic functions in more general cases?
  3. What are some practical applications of such integrals in physics or engineering?
  4. Can you derive a numerical solution to this integral?
  5. How would the result change if the bounds were modified?

Tip: Always verify substitution steps carefully to avoid errors in integration bounds and variables.

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Math Problem Analysis

Mathematical Concepts

Hyperbolic functions
Integration
Substitution
Definite integrals

Formulas

coth^2(u) = 1 + csch^2(u)
Derivative of ln(x): d/dx[ln(x)] = 1/x
Integral of coth^2(u): ∫coth^2(u) du = u - coth(u) + C

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Undergraduate Mathematics