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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
The formula for sinh(2x) cosh(2x) is:
sinh(2x) cosh(2x) = (e2x - e-2x)/2 * (e2x + e-2x)/2
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
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.
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
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.