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addmeup
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Homework Statement
Evaluate the integral
(e^-theta) cos(2theta)
I got this as my answer
e^(-theta)-sin(2theta)+cos(2theta)e^(-theta)+C
But it was wrong
All help is appreciated.
addmeup said:i got du= e^-theta
and v= sin(2theta)
addmeup said:This is probably a stupid question but why would it be?
addmeup said:ok yeah i think i got it, so now I'm at
e^-θ - sin(2θ) - ∫1/2sin(2θ) * e^-θ
what do i do with the ∫1/2sin(2θ) * e^-θ?
Take the anti derivative right?
would that be -(1/2)cos(2θ) * e^-θ?
What is the derivative of [itex]sin(2\theta)[/itex]?addmeup said:This is probably a stupid question but why would it be?
The formula for integrating e^(-theta)cos(2theta) is ∫ e^(-theta)cos(2theta) dθ = (e^(-theta)(cos(2theta) + sin(2theta)))/5 + C.
To solve the integral of e^(-theta)cos(2theta), you can use integration by parts or substitution method. For integration by parts, let u = e^(-theta) and dv = cos(2theta) dθ. For substitution method, let u = e^(-theta) and du = -e^(-theta) dθ.
The steps to integrate e^(-theta)cos(2theta) are as follows:
1. Use integration by parts or substitution method to rewrite the integral.
2. Use the chain rule to find the derivative of e^(-theta).
3. Use the double angle formula for cosine to simplify the integral.
4. Evaluate the integral and add the constant of integration.
Yes, you can use a calculator to integrate e^(-theta)cos(2theta). Most scientific calculators have an integral function that allows you to enter the function and limits of integration to solve the integral. However, some online integral calculators may not be able to handle complex functions like e^(-theta)cos(2theta).
Integrating e^(-theta)cos(2theta) has various applications in physics and engineering, such as calculating the displacement of a damped harmonic oscillator or solving for the current in a series RLC circuit. It is also used in probability and statistics to calculate the cumulative distribution function of a normal distribution.