Integration by Parts Step-by-Step Calculus Free Online Tool

Integration by Parts Calculator

Work through integration by parts step by step using \[ \int u\,dv = uv-\int v\,du. \] Enter \(u\) and \(dv\) to see how to differentiate \(u\), integrate \(dv\), and substitute the results into the formula.

Calculate Integration by Parts

Enter the parts you want to use for \(u\) and \(dv\). Common examples include x with sin(x), or x with exp(x).

Integration by Parts Formula

Integration by parts is based on the product rule for differentiation. The fundamental formula is:

\[ \boxed{ \int u\,dv = uv-\int v\,du } \]

The method transforms one integral into another. The goal is normally to choose \(u\) and \(dv\) so that the remaining integral \[ \int v\,du \] is easier to evaluate than the original integral.

How Integration by Parts Works

There are four main steps:

Choose \(u\).
Differentiate \(u\) to obtain \(du\).
Choose \(dv\) and integrate it to obtain \(v\).
Substitute everything into \(\int u\,dv=uv-\int v\,du\).

Key idea: Choose \(u\) so that differentiating it makes the expression simpler, while \(dv\) should be something that can be integrated relatively easily.

How to Choose \(u\): LIATE

A commonly taught guideline for selecting \(u\) is LIATE. It provides a useful ordering when an integral contains a product of different types of functions.

LIATE is a practical guideline rather than a mathematical requirement. Some integrals can be solved with more than one reasonable choice of \(u\) and \(dv\).

Integration by Parts Examples

Example 1: \(\int x\sin(x)\,dx\)

A common choice is:

\[ u=x, \qquad dv=\sin(x)\,dx \]

Then:

\[ du=dx, \qquad v=-\cos(x) \]

Applying the formula gives:

\[ \int x\sin(x)\,dx = -x\cos(x) + \int\cos(x)\,dx \]

\[ \int x\sin(x)\,dx = -x\cos(x)+\sin(x)+C \]

Example 2: \(\int x\cos(x)\,dx\)

Choose:

\[ u=x, \qquad dv=\cos(x)\,dx \]

Therefore:

\[ du=dx, \qquad v=\sin(x) \]

The integration-by-parts step becomes:

\[ \int x\cos(x)\,dx = x\sin(x) - \int\sin(x)\,dx \]

So:

\[ \int x\cos(x)\,dx = x\sin(x)+\cos(x)+C \]

Supported Common Expressions

The calculator recognizes a collection of common single-variable expressions when determining \(du\) and \(v\).

Note: Symbolic calculus is much broader than a fixed collection of pattern rules. For expressions that are not recognized directly, the calculator shows the appropriate derivative or integral operation instead of claiming a closed-form answer that it has not actually calculated.

When Is Integration by Parts Useful?

Integration by parts is particularly useful for integrals involving products of functions. Common examples include:

Polynomial multiplied by an exponential function.
Polynomial multiplied by a sine or cosine function.
Polynomial multiplied by a logarithm.
Logarithmic functions that become simpler after differentiation.
Inverse trigonometric functions.
Some integrals requiring repeated applications of the method.

Repeated Integration by Parts

Some integrals require integration by parts more than once. This frequently happens when a polynomial is multiplied by an exponential or trigonometric function.

For example, when integrating a product involving \(x^2\) and \(e^x\), differentiating the polynomial repeatedly eventually reduces it to zero:

\[ x^2 \rightarrow 2x \rightarrow 2 \rightarrow 0 \]

This makes repeated integration by parts a natural technique for many polynomial-exponential and polynomial-trigonometric integrals.

Integration by Parts for Definite Integrals

The same method also applies to definite integrals. Instead of adding an arbitrary constant \(C\), the formula is evaluated at the endpoints:

\[ \int_a^b u\,dv = \left[uv\right]_a^b - \int_a^b v\,du \]

If you need a numerical result for a definite integral, use the Definite Integral Calculator or the Numerical Integration Calculator .

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Integration and Differentiation

Integration by parts is directly connected to differentiation through the product rule. The product rule states:

\[ \frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx} \]

Rearranging and integrating this relationship produces the integration-by-parts formula.

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Frequently Asked Questions

What is integration by parts?

Integration by parts is a calculus technique used to transform an integral involving a product of functions. Its formula is \[ \int u\,dv=uv-\int v\,du. \]

How do I choose u and dv?

A common guideline is LIATE: logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential. In many product integrals, the function appearing earlier in this ordering is a useful candidate for \(u\).

What does dv mean?

\(dv\) represents the portion of the integral that will be integrated to obtain \(v\). Once \(v\) is found, the integration-by-parts formula can be applied.

Why is there a minus sign in the formula?

The minus sign comes from rearranging the product rule. Starting with \[ d(uv)=u\,dv+v\,du, \] rearrangement and integration produces \[ \int u\,dv=uv-\int v\,du. \]

Can integration by parts be used more than once?

Yes. Some integrals require repeated applications. Polynomial-exponential and polynomial-trigonometric integrals are common examples.

Does this calculator solve every integration-by-parts problem?

The calculator is designed around common symbolic expressions and demonstrates the integration-by-parts setup. Arbitrary symbolic integration requires a full computer algebra system and may involve techniques beyond integration by parts.

What should I use for a numerical definite integral?

Use the Definite Integral Calculator or Numerical Integration Calculator when you need a numerical value over a specified interval.

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