Integration by Parts Calculator
Evaluate integrals using ∫u dv = uv - ∫v du. Use the LIATE rule to choose u and dv. Solve x·ln(x), x·e⁶, x·sin(x) and similar integrals.
∫ Calculus📐 ∫u dv = uv - ∫v du🔢 Math
Type
Exponent n
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Integration by Parts Calculator | — | ∫u dv = uv - ∫v du | expression + C |
Step-by-Step Examples
Example 1
x·ln(x)
∫x·ln(x) dx
- u=ln(x) (L before A in LIATE)
- v=x²/2, du=1/x dx
- x²/2·ln(x)-x²/4+C
✓ x²/2·ln(x)-x²/4+C
Example 2
x·eˣ
∫x·eˣ dx
- u=x, v=eˣ
- x·eˣ-eˣ+C=(x-1)eˣ+C
✓ (x-1)eˣ+C
Example 3
x·sin(x)
∫x·sin(x) dx
- u=x, v=-cos(x)
- -x·cos(x)+sin(x)+C
✓ sin(x)-x·cos(x)+C
Real-World Applications
Calculus
Evaluates products that substitution cannot handle.
Physics
Moments of inertia and center of mass.
Engineering
Laplace and Fourier transforms.
Statistics
E[X]=∫x·f(x)dx for continuous distributions.
Common Mistakes to Avoid
⚠️
Wrong u choice
LIATE: Logs, Inverse trig, Algebraic, Trig, Exponential. Choose first type as u.
⚠️
Missing minus sign
∫u dv = uv MINUS ∫v du.
⚠️
Not repeating when needed
∫x²eˣdx needs IBP twice.
Frequently Asked Questions
What is LIATE? ▾
Choose u as: Logs, Inverse trig, Algebraic, Trig, Exponential — first type in product.
When to use IBP? ▾
Product of different types: x^n·ln(x), x^n·eˣ, x^n·sin(x).
Tabular integration? ▾
Rapid repeated IBP: alternately differentiate u and integrate dv.
Integral appears on both sides? ▾
Move to left: 2∫f dx=expr → ∫f dx=expr/2.
IBP vs substitution? ▾
Substitution: composite. IBP: product of different types.
Related Math Calculators
Formula Explorer connections
Interpretation: This relationship converts, summarizes or checks numerical quantities using standard arithmetic and measurement rules. Assumption: Use consistent units, preserve enough significant digits, and round only the final result unless the method states otherwise.