Efficient Integration of sinh(2x) cosh(2x) with Step-by-Step Homework Solution

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SUMMARY

The integral of sinh(2x) cosh(2x) dx can be solved using substitution. By letting u = sinh(2x), the differential du becomes 2 cosh(2x) dx. The integral simplifies to (1/2) ∫ u du, resulting in (sinh(2x))^2 / 4. It is crucial to include the integration constant in the final answer, which was omitted in the discussion. Verifying the solution through differentiation confirms the correctness of the integration process.

PREREQUISITES
  • Understanding of hyperbolic functions, specifically sinh and cosh.
  • Knowledge of integration techniques, particularly substitution.
  • Familiarity with differentiation to verify integration results.
  • Basic algebra skills for manipulating expressions.
NEXT STEPS
  • Study hyperbolic function properties and identities.
  • Practice integration by substitution with various functions.
  • Learn about the importance of integration constants in indefinite integrals.
  • Explore differentiation techniques to validate integration solutions.
USEFUL FOR

Students studying calculus, particularly those focusing on integration techniques, and educators seeking to enhance their teaching methods in hyperbolic functions.

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Homework Statement


int sinh(2x) cosh(2x) dx


u= sinh (2x)
du= 2 cosh (2x)

1/2 int u du

1/2 (u^2)/2

(sinh (2x))^2 / 4
 
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Looks just fine, except for one little typo where you wrote
du= 2 cosh (2x)
instead of
du= 2 cosh (2x) dx

And you forgot the integration constant :-p
 
Why don't you differentiate your answer and see if you got what you started with...
 

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