- #1
intenzxboi
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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
Efficient Integration of sinh(2x) cosh(2x) with Step-by-Step Homework Solution
- Thread starter intenzxboi
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- Check my work Integration Work
In summary, the formula for sinh(2x) cosh(2x) is (e^2x - e^-2x)/2 * (e^2x + e^-2x)/2. To efficiently integrate it, you can use the substitution method or integration by parts. To check your solution, you can differentiate the result and confirm if it matches the original function.
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
- #2
CompuChip
Science Advisor
Homework Helper
- 4,309
- 49
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
du= 2 cosh (2x)
instead of
du= 2 cosh (2x) dx
And you forgot the integration constant
- #3
NoMoreExams
- 623
- 0
Why don't you differentiate your answer and see if you got what you started with...
FAQ: Efficient Integration of sinh(2x) cosh(2x) with Step-by-Step Homework Solution
What is the formula for sinh(2x) cosh(2x)?
The formula for sinh(2x) cosh(2x) is:
sinh(2x) cosh(2x) = (e2x - e-2x)/2 * (e2x + e-2x)/2
How can I efficiently integrate sinh(2x) cosh(2x)?
The integration of sinh(2x) cosh(2x) can be done using the substitution method. Let u = 2x, then the integral becomes:
∫ sinh(u) cosh(u) du = ∫ (eu - e-u)/2 * (eu + e-u)/2 du
Using the formula for cosh(u) = (eu + e-u)/2, the integral simplifies to:
∫ sinh(u) cosh(u) du = ∫ (e2u - 1)/4 du = (e2u/8 - u/4) + C
Substituting back u = 2x, the final solution is:
∫ sinh(2x) cosh(2x) dx = (e4x/8 - x/2) + C
What is the step-by-step solution for integrating sinh(2x) cosh(2x)?
The step-by-step solution for integrating sinh(2x) cosh(2x) is as follows:
1. Use the substitution method by letting u = 2x.
2. Substitute the formula for cosh(u) = (eu + e-u)/2 in the integral.
3. Simplify the integral using the formula for sinh(u) = (eu - e-u)/2.
4. Integrate the new simplified integral.
5. Substitute back u = 2x and simplify the final solution.
Can I use any other method to integrate sinh(2x) cosh(2x)?
Yes, you can also use the integration by parts method to integrate sinh(2x) cosh(2x). Let u = sinh(2x) and dv = cosh(2x) dx. Then, the integral becomes:
∫ sinh(2x) cosh(2x) dx = sinh(2x) * (sinh(2x)/2) - ∫ (sinh(2x)/2) * (2cosh(2x)/2) dx
Using the formulas for sinh(2x) and cosh(2x), the integral simplifies to:
∫ sinh(2x) cosh(2x) dx = (sinh(2x)2/4) - ∫ (cosh(2x)/4) dx
Integrating the second term, we get:
∫ sinh(2x) cosh(2x) dx = (sinh(2x)2/4) - (sinh(2x)/8) + C
How can I check my solution for integrating sinh(2x) cosh(2x)?
You can check your solution by differentiating the result. If the derivative matches the original function, then the solution is correct. In this case, the derivative of (e4x/8 - x/2) is 2(e4x/8 - x/2) = (e4x/4 - 1), which matches the original function sinh(2x) cosh(2x). This confirms the correctness of the solution.