QuestionEvaluate the integral $\int e^{2 \theta} \sin 3 \theta d \theta$ .

Studdy Solution

STEP 1

Assumptions1. We are given the integral $\int e^{ \theta} \sin3 \theta d \theta$ . We will use the method of integration by parts, which is given by the formula $\int u dv = uv - \int v du$ .

STEP 2

First, we need to identify the functions u and dv in the integral. We choose $u = e^{2 \theta}$ and $dv = \sin \theta d \theta$ .

STEP 3

Next, we need to find the derivatives and integrals of u and dv respectively. We have $du =2e^{2 \theta} d\theta$ and $v = -\frac{1}{3}\cos3 \theta$ .

STEP 4

Now we can substitute these values into the formula for integration by parts.
$\int e^{2 \theta} \sin3 \theta d \theta = uv - \int v du$

STEP 5

Substitute the values of u, v, du into the formula.
$\int e^{2 \theta} \sin3 \theta d \theta = e^{2 \theta} \left(-\frac{1}{3}\cos3 \theta\right) - \int -\frac{1}{3}\cos3 \theta \cdot2e^{2 \theta} d\theta$

STEP 6

implify the equation.
$\int e^{2 \theta} \sin3 \theta d \theta = -\frac{1}{3}e^{2 \theta}\cos3 \theta + \frac{2}{3}\int e^{2 \theta}\cos3 \theta d\theta$

STEP 7

We have a new integral to solve, $\int e^{2 \theta}\cos3 \theta d\theta$ . We will use integration by parts again, with $u = e^{2 \theta}$ and $dv = \cos3 \theta d \theta$ . Then $du =2e^{2 \theta} d\theta$ and $v = \frac{1}{3}\sin3 \theta$ .

STEP 8

Substitute these values into the formula for integration by parts.
$\int e^{2 \theta}\cos3 \theta d\theta = uv - \int v du$

STEP 9

Substitute the values of u, v, du into the formula.
$\int e^{2 \theta}\cos3 \theta d\theta = e^{2 \theta} \left(\frac{}{3}\sin3 \theta\right) - \int \frac{}{3}\sin3 \theta \cdot2e^{2 \theta} d\theta$

STEP 10

implify the equation.
$\int e^{2 \theta}\cos3 \theta d\theta = \frac{}{3}e^{2 \theta}\sin3 \theta - \frac{2}{3}\int e^{2 \theta}\sin3 \theta d\theta$

STEP 11

Substitute this result back into the equation from6.
$\int e^{ \theta} \sin3 \theta d \theta = -\frac{}{3}e^{ \theta}\cos3 \theta + \frac{}{3}\left(\frac{}{3}e^{ \theta}\sin3 \theta - \frac{}{3}\int e^{ \theta}\sin3 \theta d\theta\right)$

STEP 12

implify the equation.
$\int e^{2 \theta} \sin \theta d \theta = -\frac{}{}e^{2 \theta}\cos \theta + \frac{2}{9}e^{2 \theta}\sin \theta - \frac{4}{9}\int e^{2 \theta}\sin \theta d\theta$

STEP 13

We notice that the integral we are trying to solve has appeared again on the right side of the equation. We can move it to the left side to solve for it.
$\int e^{2 \theta} \sin3 \theta d \theta + \frac{}{9}\int e^{2 \theta}\sin3 \theta d\theta = -\frac{}{3}e^{2 \theta}\cos3 \theta + \frac{2}{9}e^{2 \theta}\sin3 \theta$

STEP 14

Combine the integrals on the left side.
$\left( + \frac{4}{9}\right)\int e^{2 \theta} \sin3 \theta d\theta = -\frac{}{3}e^{2 \theta}\cos3 \theta + \frac{2}{9}e^{2 \theta}\sin3 \theta$

STEP 15

implify the equation.
$\frac{13}{9}\int e^{2 \theta} \sin3 \theta d\theta = -\frac{}{3}e^{2 \theta}\cos3 \theta + \frac{2}{9}e^{2 \theta}\sin3 \theta$

STEP 16

Finally, we can solve for the original integral by multiplying both sides of the equation by $\frac{9}{13}$ .
$\int e^{2 \theta} \sin3 \theta d\theta = \frac{9}{13}\left(-\frac{}{3}e^{2 \theta}\cos3 \theta + \frac{2}{9}e^{2 \theta}\sin3 \theta\right)$

STEP 17

implify the equation to get the final answer.
$\int e^{2 \theta} \sin3 \theta d\theta = -\frac{3}{13}e^{2 \theta}\cos3 \theta + \frac{2}{13}e^{2 \theta}\sin3 \theta + C$ Where C is the constant of integration.
So, the solution to the integral $\int e^{2 \theta} \sin3 \theta d \theta$ is $-\frac{3}{13}e^{2 \theta}\cos3 \theta + \frac{2}{13}e^{2 \theta}\sin3 \theta + C$ .

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