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This integration by parts calculator solves integrals of products, like \(x e^x\), \(x^2\sin x\), and \(x\ln x\). It shows the LIATE choice, writes out \(u\), \(du\), \(dv\), and \(v\), and applies the formula. If parts is needed more than once, you see every round.
How to Use It
- Type the product, for example
x^2 e^(2x). - Press Solve.
- Open the Integration by Parts step to see \(u\), \(dv\), and how \(v\) was found.
The Integration by Parts Formula
\[\int u\,dv = uv – \int v\,du\]
It comes from the product rule for derivatives. You split the integrand into two pieces: \(u\) (which you will differentiate) and \(dv\) (which you will integrate). A good split makes \(\int v\,du\) easier than the integral you started with. For a deeper lesson, read integration by parts: when to use it and how it works.
How to Choose u: The LIATE Rule
Pick \(u\) as the function that comes first in this list:
| Letter | Type | Examples |
|---|---|---|
| L | Logarithmic | \(\ln x\), \(\log_{10} x\) |
| I | Inverse trig | \(\tan^{-1} x\), \(\sin^{-1} x\) |
| A | Algebraic | \(x\), \(x^2\), \(3x + 1\) |
| T | Trigonometric | \(\sin x\), \(\cos 3x\) |
| E | Exponential | \(e^x\), \(2^x\) |
Everything else becomes \(dv\). For \(x\cos x\), “A” comes before “T”, so \(u = x\) and \(dv = \cos x\,dx\).
Worked Examples
Example 1: \(\int x e^x\,dx\)
\(u = x\), \(du = dx\). \(dv = e^x\,dx\), \(v = e^x\).
\(\int xe^x\,dx = xe^x – \int e^x\,dx = xe^x – e^x + C\).
Example 2: \(\int x\ln x\,dx\)
“L” comes first, so \(u = \ln x\), \(du = \frac1x\,dx\). \(dv = x\,dx\), \(v = \frac{x^2}{2}\).
\(\frac{x^2}{2}\ln x – \int \frac{x^2}{2}\cdot\frac1x\,dx = \frac{x^2}{2}\ln x – \frac{x^2}{4} + C\).
Example 3: \(\int x^2\sin x\,dx\) (parts twice)
Round 1: \(u = x^2\), \(dv = \sin x\,dx\), so \(v = -\cos x\). This gives \(-x^2\cos x + \int 2x\cos x\,dx\).
Round 2: \(u = 2x\), \(dv = \cos x\,dx\). This gives \(2x\sin x – \int 2\sin x\,dx = 2x\sin x + 2\cos x\).
Answer: \(-x^2\cos x + 2x\sin x + 2\cos x + C\).
Example 4: \(\int e^x\sin x\,dx\) (the loop trick)
Using parts twice brings back the original integral. Call it \(I\). You get \(I = e^x\sin x – e^x\cos x – I\). Solve for \(I\):
\(I = \frac{e^x(\sin x – \cos x)}{2} + C\).
The Tabular Method (Shortcut)
When \(u\) is a polynomial, you can use a table. Write derivatives of \(u\) in one column until you reach 0. Write integrals of \(dv\) in the other column. Multiply along diagonals with signs \(+, -, +, \ldots\). For \(\int x^3 e^x\,dx\) this gives \(e^x(x^3 – 3x^2 + 6x – 6) + C\) in one line.
Common Mistakes
- Choosing u = eˣ. Then \(\int v\,du\) gets harder, not easier. Follow LIATE.
- Sign errors with sine and cosine. \(\int \sin x\,dx = -\cos x\). Watch the minus.
- Stopping after one round. \(x^2\) needs two rounds, \(x^3\) needs three.
- Using parts when substitution works. \(\int x\cos(x^2)\,dx\) is a u-substitution problem.
Learn the Math Behind the Integration by Parts Calculator
Integration by parts is the reverse product rule. Use it for products like x e^x, x sin x or ln x, and choose u with LIATE.
Frequently Asked Questions
When should I use integration by parts?
Use it for a product of two different kinds of functions, like a polynomial times an exponential, or a log by itself (\(\int \ln x\,dx\)).
Why is \(\int \ln x\,dx\) done by parts?
Write it as \(\int \ln x \cdot 1\,dx\). Let \(u = \ln x\) and \(dv = dx\). The answer is \(x\ln x – x + C\).
Does the order of u and dv matter?
Yes. A bad choice still follows the formula, but leads to a harder integral. LIATE gives a good choice almost every time.