Integration by Parts Explained

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1. What Is Integration by Parts?

2. Where Does Integration by Parts Come From?

3. How To Integrate by Parts

4. Pro-Tip

5. Integration by Parts Examples

What Is Integration by Parts?

Integration by parts is the most useful integration technique for evaluating the integral of a product of functions. (If you need to review integrals in general, look at this guide to integrals).

After separating a single function into a product of two functions, we can easily evaluate the function's integral by applying the integration by parts formula:

In this formula, $du$ represents the derivative of $u$, while $v$ represents the integral of $dv$. The integral of the product of $u$ and $v$ is then evaluated in terms of the integral of the product of $du$ and $v$. It's often much easier to solve the integral that results from this formula than it is to directly evaluate the original single function. Integration by parts gives us an efficient way to integrate functions that would otherwise be very difficult to integrate, and it's particularly helpful in tricky cases such as inverse trigonometric functions and logarithmic functions. 

Where Does Integration by Parts Come From?

You might be surprised to learn that the formula for integration by parts is derived directly from the familiar Product Rule theorem for derivatives. Let's review the formula for the Product Rule, which states that the derivative of a product of two functions is equal to the first function times the derivative of the second, plus the second function times the derivative of the first:

Multiplying both sides of the equation by the differential $dx$, we obtain:

Now, let's take the indefinite integral of both sides. Remember that the integral of a sum is simply equal to the sum of their integrals. Finding the antiderivative of both sides gives us the following:

All that's left to do now is rearrange the equation! Let's subtract $\int vdu$ from both sides. Does the resulting equation look familiar?

In this way, we can use the Product Rule to obtain the formula for integration by parts. So, if you ever forget the formula for integration by parts during a test, you can rely on the Product Rule to help rediscover it!

How To Integrate by Parts

Let's outline some simple steps for how to integrate by parts:

1. Choose $u$ and $dv$ to separate the integrand into a product of functions.

2. Differentiate $u$ to find $du$, and integrate $dv$ to find $v$.

3. Plug $u$, $v$, $du$, and $dv$ into the integration by parts formula: $\int udv = uv - \int vdu$

4. Solve, and simplify where needed.

Let's do one example together in greater detail. Suppose we want to evaluate $\int xe^xdx$. First, we separate the function into a product of two functions; one is a polynomial and one is an exponential function.

To do this, we need to designate one expression as $u$ and one as $dv$. Let's choose $u = x$. Since we've chosen $u$, what's left over must determine $dv$. Then $dv = e^xdx$.

Our next step is to differentiate $u$ to find $du$. It's clear that $\frac{du}{dx}=\frac{dx}{dx}=1$. Then, multiplying both sides by the differential $dx$, we find that $du = 1dx$. Now, let's integrate $dv$ to find $v$. Since the integral of $e^x$ is $e^x$ itself, we have $v = e^x$. Finally, we plug $u = x$, $v = e^x$, $du = 1dx$, and $dv = e^xdx$ into the integration by parts formula. We have:

$\int udv = uv - \int vdu$

$\int xe^xdx = xe^x - \int e^x(1dx)$

$\int xe^xdx = xe^x - e^x + C$

We've found that $\int xe^xdx = xe^x - e^x + C$. Remember to include the constant of integration in your final answer!

LIATE

Now, the question we have to ask is: how do we know what to set equal to $u$ each time, and what to set equal to $dv$? This decision is where the difficulty lies for many students. If you have a strong intuition for what's easiest to integrate and what's easiest to differentiate, you can denote $u$ as the term you'd rather differentiate and $dv$ as the term you'd rather integrate.

If you feel unsure about what would be easiest to differentiate and integrate, the rule of thumb is to use the mnemonic “LIATE,” an acronym based on an ordered list of different function types. This acronym can help students identify when to use integration by parts. Usually, the higher a function sits on the list, the higher precedence it takes as $u$. With this logic, LIATE can help guide your decision on which term to designate as $u$:

• L stands for logarithmic functions: such as $\ln{x}$, $\log_3(x)$, etc.

• I stands for inverse trigonometric functions: such as $\arcsin(x)$, $\arctan(x)$, etc.

• A stands for algebraic functions: such as $x$, $x^2$, $4x^3$, etc.

• T stands for trigonometric functions: such as $\sin(x)$, $cos(x)$, etc.

• E stands for exponential functions: such as $e^x$, $4^x$, etc.

See if you recognize any of these five function types inside the integral that you are trying to evaluate. Designate $u$ as the term that is categorized highest on LIATE. Then, you can set $dv$ equal to whatever is left over.

Pro-Tip

To remember the formula for integration by parts, it might be helpful to use another mnemonic device. One popular choice is “ultraviolet voodoo,” where “ultraviolet” corresponds to $uv$ and “voodoo” corresponds to $\int vdu$.

A mnemonic for Spanish speakers is, “Un día vi una vaca vestida de uniforme.” With this device, un ( $u$ ) día vi ( $dv$ ) represents the left-side of the integration by parts formula, and una ( $u$ ) vaca ( $v$ ) vestida ( $v$ ) de uniforme $(du)$ represents the right-hand side. Can you think of any other mnemonic devices?

Integration by Parts Examples

Let's work through the following integrals together.

1. Evaluate $\int x\sin(x)dx$.

• Let $u=x$ and $dv=\sin(x)dx$.

• Differentiating $u$, we have $du = 1dx$.

• Integrating $dv$, we find $v = - \cos(x)$.

• Now, let's plug $u=x$, $v = - \cos(x)$, $du = 1dx$, and $dv=\sin(x)dx$ into the integration by parts formula.

  $\int udv = uv - \int vdu$

  $\int x\sin(x)dx = -x \cos(x) - \int - \cos(x)dx$

  $\int x\sin(x)dx = -x \cos(x) + \sin(x) + C$

2. Evaluate $\int x^2 \sin(x)dx$.

• Let $u = x^2$ and $dv = \sin{(x)}dx$.

• Differentiating $u$, we have $du = 2xdx$.

• Integrating $dv$, we have $v = - \cos{(x})$.

• Now, let's plug $u = x^2$, $v = - \cos{(x})$, $du = 2xdx$, and $dv = \sin{(x)}dx$ into the integration by parts formula.

  $\int udv = uv - \int vdu$

  $\int x^2 \sin(x)dx = x^2 (-\cos{(x)}) - \int (- \cos{(x)})2xdx$

  $\int x^2 \sin(x)dx = -x^2\cos{(x)} + 2 \int x \cos{(x)}dx$

Sometimes, you'll have to integrate by parts more than once in the same problem. In this case, we'll need to use it again to evaluate $\int x \cos{(x)}dx$.

• Let $u_1 = x$ and $dv_1 = \cos{(x)}dx$.

• Differentiating $u_1$, we have $du_1 = 1dx$.

• Integrating $dv_1$, we have $v_1 = \sin{(x)}$.

• Now, we plug $u_1 = x$, $v_1 = \sin{(x)}$, $du_1 = 1dx$, and $dv_1 = \cos{(x)}dx$ into the integration by parts formula.

  $\int udv = uv - \int vdu$

  $\int x \cos{(x)}dx = x \sin{(x)} - \int \sin{(x})dx$

  $\int x \cos{(x)}dx = x \sin{(x)} -(- \cos{(x)})$

  $\int x \cos{(x)}dx = x \sin{(x)} + \cos{(x)} + C$

Now, we've found that $\int x \cos{(x)}dx = x \sin{(x)} + \cos{(x)} + C$. We can use substitution to plug this value into our original integral formula to finish the problem. Note that since $C$ is arbitrary, we don't have to multiply it by 2 when distributing. Rearranging our equation, we have:

  $\int x^2 \sin(x)dx = -x^2\cos{(x)} + 2 \int x \cos{(x)}dx$

  $\int x^2 \sin(x)dx = -x^2\cos{(x)} + 2(x \sin{(x)} + \cos{(x)}$

  $\int x^2 \sin(x)dx = -x^2\cos{(x)} + 2x \sin{(x)} + 2 \cos{(x)} + C$

As you can see, integration by parts is a valuable instrument in your mathematical toolbox.

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