Solved example of integration by parts
We can solve the integral $\int x\cos\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
The derivative of the linear function is equal to $1$
First, identify $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$
Now replace the values of $u$, $du$ and $v$ in the last formula
Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$
Any expression multiplied by $1$ is equal to itself
The integral $-\int\sin\left(x\right)dx$ results in: $\cos\left(x\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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