The integration by parts guide walks through the formula ∫u dv = uv − ∫v du with the LIATE rule for choosing u. Select from 8 worked examples to see full step-by-step solutions for common integral types.
Integration by Parts Guide
Formula: ∫u dv = uv − ∫v du
LIATE rule for choosing u: Logarithmic → Inverse trig → Algebraic → Trigonometric → Exponential
Integral
Step-by-Step Solution
Answer
How to Use the Integration by Parts Guide
Integration by parts converts products of functions into simpler integrals. The formula ∫u dv = uv − ∫v du comes from integrating the product rule: d(uv)/dx = u·v' + v·u'.
The LIATE Rule
When integrating a product, choose u as the type appearing earliest in LIATE: Logarithmic functions (ln, log), Inverse trig functions (arctan, arcsin), Algebraic functions (x, x², polynomials), Trigonometric functions (sin, cos), Exponential functions (eˣ, aˣ). The remaining factor becomes dv.
Worked Example: ∫x·sin(x) dx
x is Algebraic, sin(x) is Trigonometric. LIATE says choose u = x (A before T). Then dv = sin(x)dx, so du = dx and v = −cos(x). Apply formula: ∫x·sin(x)dx = x·(−cos(x)) − ∫(−cos(x))dx = −x·cos(x) + sin(x) + C.
Circular Integration
For ∫eˣ·cos(x)dx, applying IBP twice brings back the original integral. Let I = ∫eˣ·cos(x)dx. After two applications: I = eˣ·sin(x) + eˣ·cos(x) − I. Solving: 2I = eˣ(sin(x)+cos(x)), so I = eˣ(sin(x)+cos(x))/2 + C.
FAQ
What is integration by parts?
Integration by parts is a technique based on the product rule for differentiation: ∫u dv = uv − ∫v du. It converts an integral of a product into (usually) a simpler integral. The formula follows from d(uv)/dx = u(dv/dx) + v(du/dx).
What is the LIATE rule?
LIATE is a mnemonic for choosing u (the part to differentiate): Logarithmic, Inverse trigonometric, Algebraic (polynomials), Trigonometric, Exponential. Choose u to be the type that appears earliest in LIATE — it tends to simplify after differentiation.
When do I use integration by parts?
Use integration by parts when the integrand is a product of functions from different categories: x·sin(x), x²·eˣ, x·ln(x), eˣ·cos(x), x·arctan(x). It also works for single functions like ln(x) or arcsin(x) by setting dv = dx.
Is this guide free?
Yes, completely free with no signup required. All examples run in your browser.
What if integration by parts leads to another integration by parts?
Sometimes you need to apply integration by parts twice. For ∫eˣ·cos(x)dx, applying twice brings back the original integral — rearrange algebraically to solve. This 'circular' method gives the answer without infinite steps.