**Integration by parts** is a method to calculate indefinite integrals by using the differential of the product of two functions.

If we have two functions, *u* and *v*, the differential of their product will be:

If we integrate both sides of the mathematical expression, we’ll get:

\[\int d(u \cdot v)=\int \left (u \cdot dv + v \cdot du \right ) \tag{2}\]We know that the integral of the derivative of a function gives the function itself. So:

\[\int d(u \cdot v)= u \cdot v \tag{3}\]Also, the integral of the sum of two functions is the sum of the integral of each function:

\[\int \left (u \cdot dv + v \cdot du \right ) = \int u \cdot dv + \int v \cdot du \tag{4}\]Replacing equations (3) and (4) in equation (2) we get:

\[ u \cdot v =\int u \cdot dv + \int v \cdot du \tag{5} \]Rearranging equation (5) gives the mathematical expression for **integration by parts**:

**Example 1**. Solve the following integral using integration by parts:

As we can see, the integral contains the product of two functions: *x* and *cos(x)*. We are going to replace these with *u* and *v*.

**Step 1**. Replace the functions with *u* and *v*

First we replace:

\[u = x \tag{7}\]Applying **differentiation** to equation (7) we get:

Second we replace:

\[dv = cos(x)dx \tag{9}\]Applying **integration** to equation (9) we get:

**Step 2**. Use general equation (6) and replace *u* and *v* with calculated functions:

**Step 3**. Solve the right side of the equation

We know that:

\[\int sin(x) dx = – cos(x) + C \tag{13}\]Replacing the result of the integral (13) in equation (12) we get the result of our integration by parts:

\[\int x \cdot cos(x) dx = x \cdot sin(x) + cos(x) + C\]where *C* is a constant of integration.

**Example 2**. Solve the following integral using integration by parts:

As we can see, the integral contains the product of two functions: *x* and *e ^{x}*. We are going to replace these with

*u*and

*v*.

**Step 1**. Replace the functions with *u* and *v*

First we replace:

\[u = x \tag{14}\]Applying **differentiation** to equation (14) we get:

Second we replace:

\[dv = e^x dx \tag{16}\]Applying **integration** to equation (16) we get:

**Step 2**. Use general equation (6) and replace *u* and *v* with calculated functions:

**Step 3**. Solve the right side of the equation

We know that:

\[\int e^x dx = e^x + C \tag{20}\]Replacing the result of the integral (20) in equation (19) we get the result of our integration by parts:

\[\int x \cdot e^x dx = x \cdot e^x – e^x + C\]which in simplified form is:

\[\int x \cdot e^x dx = e^x \cdot (x-1) + C\]where *C* is a constant of integration.

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