# Integrate e^(-theta)cos(2theta): Get Help Now!

In summary: If there are multiple integrals involved, things get a little more complicated, but the basic idea is the same.) In summary, this person got their answer for the integral wrong, and they need help figuring out what the derivative is.

## 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.

well you don't have "1/2" in there anywhere

u=e so du=?

dv= cos2θ dθ so v= ?

i got du= e^-theta
and v= sin(2theta)

i got du= e^-theta
and v= sin(2theta)

v would be "1/2 sin2θ", right?

This is probably a stupid question but why would it be?

This is probably a stupid question but why would it be?

$$\int cos n\theta d\theta$$

let t = nθ ⇒ dt = ndθ or dt/n = dθ (n is a constant)

$$\therefore \int cos n\theta d\theta \equiv \int \frac{cos t}{n} dt = \frac{1}{n} sin t=\frac{1}{n}sin(n\theta)$$

see where the 1/2 comes from?

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?

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?

integrate by parts again

This is probably a stupid question but why would it be?
What is the derivative of $sin(2\theta)$?

In general, if $\int f(x)dx= F(x)+ C$ then to integrate $\int f(ax+b) dx$, let u= ax+ b so that du= a dx or (1/a)du= dx. The integral becomes $(1/a)\int f(u)du= (1/a)F(u)+ C= (1/a)F(ax+ b)+ C$.

That is, for f(ax+b), just as, if you were differentiating, you would have to multiply by a (by the chain rule), so when integrating, you divide by a.

(That works for a simple linear substitution.

## 1. What is the formula for integrating e^(-theta)cos(2theta)?

The formula for integrating e^(-theta)cos(2theta) is ∫ e^(-theta)cos(2theta) dθ = (e^(-theta)(cos(2theta) + sin(2theta)))/5 + C.

## 2. How do I solve the integral of e^(-theta)cos(2theta)?

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.

## 3. What are the steps to integrate e^(-theta)cos(2theta)?

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.

## 4. Can I use a calculator to integrate e^(-theta)cos(2theta)?

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).

## 5. What are some applications of integrating 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.

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