Math Problem Statement
Solution
To solve the integral
let's proceed step by step:
Step 1: Simplify the integral
The hyperbolic cotangent squared function is defined as:
The integral becomes:
Step 2: Change of variables
Let . Then, the derivative is:
Substituting this, the integral becomes:
Step 3: Integrate
Recall that: where is the hyperbolic cosecant squared. Thus:
From hyperbolic integrals, we know:
Hence:
Step 4: Back-substitute
Returning to , we have:
Step 5: Evaluate the bounds
At :
At :
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:
Final result:
Would you like further clarification or a numerical evaluation of this result?
Related Questions:
- What are the steps for deriving the hyperbolic trigonometric identities used here?
- How do we handle integrals involving hyperbolic functions in more general cases?
- What are some practical applications of such integrals in physics or engineering?
- Can you derive a numerical solution to this integral?
- 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