# Step-by-step Solution

## Trigonometric integral $\int e^t\cos\left(t\right)dt$

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asin
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sinh
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### Videos

$\frac{1}{2}\left(e^t\cos\left(t\right)+e^t\sin\left(t\right)\right)+C_0$

## Step-by-step explanation

Problem to solve:

$\int\left(e^t\cos\left(t\right)\right)dt$
1

Use the integration by parts theorem to calculate the integral $\int e^t\cos\left(t\right)dt$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
2

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=e^t}\\ \displaystyle{du=e^tdt}\end{matrix}$

$\frac{1}{2}\left(e^t\cos\left(t\right)+e^t\sin\left(t\right)\right)+C_0$
$\int\left(e^t\cos\left(t\right)\right)dt$

### Main topic:

Integration by parts

~ 1.61 seconds

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