Free integration by parts calculator with step-by-step solutions. Master the integration by parts formula using LIATE rule. Practice integration by parts examples, solve practice problems, and learn when to use integration by parts. Complete guide to integration by parts equation with proof and tabular method for definite integrals.
Enter any function requiring integration by parts, such as x*exp(x), x*sin(x), ln(x), or other products of functions
Get automatic function analysis using the LIATE rule to determine optimal u and dv choices for integration by parts
View detailed step-by-step integration by parts solutions with complete mathematical reasoning and calculations
Practice integration by parts with examples and questions to master different integration techniques and concepts
Start with simple integration by parts examples like ∫x·e^x dx. Master the LIATE rule: Logarithmic (L), Inverse trigonometric (I), Algebraic (A), Trigonometric (T), Exponential (E) priority order for choosing u and dv.
Integration by parts is a fundamental calculus technique for integrating products of two functions. When the integrand is a product of functions, direct integration is often difficult, making the integration by parts rule essential for solving complex integrals.
This integration by parts equation shows that u and v are functions of the integration variable, while du and dv are their respective differentials.
The LIATE method helps determine which function to choose as u and which as dv in integration by parts:
Choose functions higher in the LIATE order as u, and functions lower in the order as dv for optimal integration by parts solutions.
Let's understand integration by parts through this classic example:
Step 1: Choose u = x (algebraic function), dv = e^x dx (exponential function)
Step 2: Calculate du = dx, v = e^x
Step 3: Apply integration by parts formula: ∫x e^x dx = x e^x - ∫e^x dx
Step 4: Solve: ∫x e^x dx = x e^x - e^x + C = e^x(x-1) + C
Integration by parts is particularly effective in these scenarios:
ln(x), log(x)
arcsin(x), arctan(x)
x, x², x³
sin(x), cos(x)
eˣ, 2ˣ
Choose functions with higher priority as u and lower priority as dv. This strategy ensures du becomes simpler than u, while v doesn't become more complex than dv. Example: In ∫x·sin(x) dx, x is algebraic (A) and sin(x) is trigonometric (T), so by LIATE rule choose u=x, dv=sin(x)dx.
LIATE Analysis: x (algebraic) vs cos(x) (trigonometric) → Choose u=x, dv=cos(x)dx
Calculate: du=dx, v=sin(x)
Solution: ∫x cos(x) dx = x sin(x) - ∫sin(x) dx = x sin(x) + cos(x) + C
Technique: Rewrite as ∫ln(x)·1 dx
LIATE Analysis: ln(x) (logarithmic) vs 1 (algebraic) → Choose u=ln(x), dv=dx
Calculate: du=1/x dx, v=x
Solution: ∫ln(x) dx = x ln(x) - ∫x·(1/x) dx = x ln(x) - x + C
Multiple integration by parts required:
First iteration: u=x², dv=eˣdx → ∫x² eˣ dx = x² eˣ - ∫2x eˣ dx
Second iteration: u=2x, dv=eˣdx → ∫2x eˣ dx = 2x eˣ - ∫2 eˣ dx
Final solution: ∫x² eˣ dx = eˣ(x² - 2x + 2) + C
Cyclic integration technique:
Let I = ∫eˣ sin(x) dx, after two integration by parts:
I = eˣ sin(x) - eˣ cos(x) - I
Solving: I = ½ eˣ(sin(x) - cos(x)) + C
Mastering integration by parts requires extensive practice. Start with simple single integration by parts examples, then progress to complex problems requiring multiple integrations. Remember the LIATE rule, but apply it flexibly - sometimes specific situations require strategy adjustments for optimal solutions.
When multiple integration by parts is required, the tabular method dramatically simplifies calculations. Particularly effective for products of polynomials with exponential or trigonometric functions.
| Differentiate | Sign | Integrate |
|---|---|---|
| x³ | + | eˣ |
| 3x² | - | eˣ |
| 6x | + | eˣ |
| 6 | - | eˣ |
| 0 | eˣ |
Result: ∫x³ eˣ dx = eˣ(x³ - 3x² + 6x - 6) + C
Integration by parts applies equally to definite integrals, with careful attention to limit evaluation.
Choose: u = x, dv = eˣ dx
Then: du = dx, v = eˣ
Apply formula:
∫₀¹ x eˣ dx = [x eˣ]₀¹ - ∫₀¹ eˣ dx
= [x eˣ]₀¹ - [eˣ]₀¹
= (1·e¹ - 0·e⁰) - (e¹ - e⁰)
= e - (e - 1) = 1