# Math virtual assistant

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# Step-by-step Solution

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## Answer

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

## Step-by-step explanation

Problem to solve:

$\int\:e^{x\:}cos\left(x\right)dx$
1

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

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

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=e^x}\\ \displaystyle{du=e^xdx}\end{matrix}$

## Answer

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

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$\int\:e^{x\:}cos\left(x\right)dx$

### Main topic:

Integration by parts

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