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Calculate the definite integral of x/(cosh(x))^n from 0 to infinity.
Published on December 5, 2024
Calculus
The integral \int_{0}^{\infty} \frac{x}{(\cosh x)^2} dx evaluates to ln(2).
Question
Integration from zero to infinity of x/(coshx)^n
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Published on December 5, 2024
The integral \int_{0}^{\infty} \frac{x}{(\cosh x)^2} dx evaluates to ln(2).
Integration from zero to infinity of x/(coshx)^n
To evaluate the limit, direct substitution results in an indeterminate form (0/0). Factoring the numerator reveals a common factor of (x-3), which cancels with the denominator. Substituting x=3 into the simplified expression yields a final answer of 58.
The integrals were evaluated using trigonometric substitution, partial fraction decomposition, and substitution. Integral c) was solved using trigonometric substitution, resulting in x/(√(1-x^2)) + C. Integral d) diverged to infinity. Integral e) was solved using partial fraction decomposition, leading to a solution involving natural logarithms and a constant term.
The inverse Laplace transform of 1/x^(3/2) is 2/(sqrt(pi))*sqrt(x).
The Laplace transform of sin^2(t) is 2 / (s * (s^2 + 4)).
The Laplace transform of e^(t/2) is 1/(s - 1/2), provided s > 1/2.