integral from 0 to pi ((e^cos(t)) sin(2t))dt
let x = cost
then dx = -sint dt
Now you have
∫ e^cost sin2t dt = ∫e^cost * 2sint cost dt
= ∫-2x e^x dx
Do that using integration by parts.
To find the integral of the function \(f(t) = e^{\cos(t)} \sin(2t)\) from \( t = 0\) to \(t = \pi\), we can use the method of integration by substitution. Let's follow these steps to evaluate the integral:
Step 1: Find the derivative of the function inside the integral.
The derivative of \(\cos(t)\) with respect to \(t\) is \(-\sin(t)\). Hence, \(d(\cos(t)) = -\sin(t)dt\).
Step 2: Determine the appropriate substitution.
In this case, we observe that the integrand \(e^{\cos(t)} \sin(2t)\) includes the term \(\cos(t)\), which is the derivative of \(\sin(t)\). Therefore, we can set \(u = \sin(t)\) as our substitution.
Step 3: Compute the values of the new variables.
Using the substitution \(u = \sin(t)\), we derive \(du = \cos(t)dt\).
Step 4: Rewrite the integral with the new variables.
Now we replace the terms in the integral with the new variables:
\(\int_{0}^{\pi} e^{\cos(t)} \sin(2t)dt = \int_{u(0)}^{u(\pi)} e^{u} \sin(2t)dt\)
Since \(u = \sin(t)\), we need to express \(\sin(2t)\) in terms of \(u\). Recall the double angle identity: \(2\sin(t)\cos(t) = \sin(2t)\).
To express \(\sin(2t)\) in terms of \(u\), we use the substitution \(u = \sin(t)\) and rewrite \(\cos(t)\) as \(\sqrt{1 - u^2}\). Therefore, \(2\sin(t)\cos(t) = 2u\sqrt{1 - u^2}\).
Our integral now becomes:
\(\int_{u(0)}^{u(\pi)}e^{u}(2u\sqrt{1 - u^2})du\)
Step 5: Evaluate the integral.
To evaluate this integral, we should notice that there is no elementary function that can be used to get the antiderivative of \(e^{u}(u\sqrt{1 - u^2})\). Therefore, we need to use numerical methods or approximation techniques to find the value of the integral.