Calculus

Integration by Parts Calculator

Evaluate derivatives, integrals, series approximations, and functional transforms for Integration by Parts with complete mathematical derivations.

Reviewed by Sanjay Samanta & Academic Review Board
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Last updated: August 2026
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Verified Mathematical Solution

Enter Components

Integration by Parts Formula: $$ \int u dv = uv - \int v du $$

Result: Indefinite Integral

Step-by-step Visualization

u:

du:

dv:

v = ∫dv:

Apply Integration by Parts Formula: $$ \int u dv = uv - \int v du $$

uv:

∫vdu:

Therefore, $$ \int u dv = uv - \int v du = $$

Direct Answer & Overview
Verified Educational Guide

How to Calculate Integration by Parts

Evaluate derivatives, integrals, series approximations, and functional transforms for Integration by Parts with complete mathematical derivations.

Primary Mathematical Formula Standard Mathematical Model
Standard Equation
ƒ(x)
Q.E.D.
f(x)=∫g(x) dxordfdxf(x) = \int g(x)\,dx \quad \text{or} \quad \frac{df}{dx}
Evaluated with exact mathematical formulation • Rigorously verified
Exact Formula
Input Parameters
Required
1
u (differentiable function): Value for u (differentiable function)
2
dv (integrable function): Value for dv (integrable function)
Expected Outputs
Calculated
Computed Integration by Parts Calculator result with step-by-step mathematical breakdown
Exact numerical value and verified formula evaluation
Worked Numerical Example
Instant Verification
Calculate the exact sum and simplified product of fractions a/b = 3/4 and c/d = 2/5.
→ Find the least common multiple of denominators 4 and 5: LCM(4, 5) = 20.; Multiply (3/4) by (5/5) to get 15/20. Multiply (2/5) by (4/4) to get 8/20.
Sum = 23/20 (1.15 as decimal, 1 3/20 as mixed fraction)

What Is the Integration by Parts Calculator?

Evaluate derivatives, integrals, series approximations, and functional transforms for Integration by Parts with complete mathematical derivations.

Understanding Integration by Parts

Integration by Parts is a powerful technique used to integrate products of functions. It's particularly useful when dealing with integrals of the form ∫f(x)g'(x)dx, where direct integration is challenging. The formula, derived from the product rule of differentiation, is $$ \int u dv = uv - \int v du $$. To apply it, you must choose which part of your integrand will be 'u' and which will be 'dv'. A helpful guideline is the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) for choosing 'u'. Remember, the goal is to choose 'u' and 'dv' such that ∫vdu is simpler than the original integral ∫udv.

For further learning, you can explore resources like:

How to Use the Integration by Parts Calculator

Using this calculator is straightforward. Enter your known values into the fields below, and the solver will compute the result immediately:

• u (differentiable function)

Example input: 0.

• dv (integrable function)

Example input: 0.

Formula Reference
\(f(x) = \int g(x)\,dx \quad \text{or} \quad \frac{df}{dx}\)

Sample Problem: Solving Operations with Fractions

Worked Example
Problem Statement

Calculate the exact sum and simplified product of fractions a/b = 3/4 and c/d = 2/5.

1

Determine the Least Common Denominator (LCD)

Find the least common multiple of denominators 4 and 5: LCM(4, 5) = 20.

\text{LCD} = 20
2

Convert Both Fractions to Equivalent Fractions

Multiply (3/4) by (5/5) to get 15/20. Multiply (2/5) by (4/4) to get 8/20.

\frac{3}{4} = \frac{15}{20}, \quad \frac{2}{5} = \frac{8}{20}
3

Add Numerators Over Common Denominator

Add 15 + 8 = 23 over denominator 20: Result = 23/20 = 1 3/20 = 1.15.

\frac{15 + 8}{20} = \frac{23}{20} = 1\frac{3}{20}
Final Result Sum = 23/20 (1.15 as decimal, 1 3/20 as mixed fraction)

How to Calculate Integration by Parts Step-by-Step

Understanding the underlying solution workflow helps build mathematical intuition and independently verify results:

1
Identify and note down the given values for: u (differentiable function), dv (integrable function).
2
Set up the primary formula: \(f(x) = \int g(x)\,dx \quad \text{or} \quad \frac{df}{dx}\). Substitute the identified values into their respective positions.
3
Perform integration, differentiation, or limit evaluation on the mathematical function.
4
Round the final calculated answer to the required decimal accuracy or significant figures.

Real-World Applications of Integration by Parts Calculator

Practical scenarios where integration by parts calculator calculations are applied across engineering, business, and everyday problem solving:

Culinary Recipe & Industrial Food Batching

Chefs and commercial bakers scale fractional ingredient proportions (e.g. 3/4 cup, 2 1/3 lbs) up or down without altering moisture ratios.

Carpentry & Construction Fractional Framing

Woodworkers use tape measures calibrated in 1/16th and 1/32nd-inch increments, requiring instant addition and subtraction of mixed fractions.

Precision Mechanical Machining Tolerances

CNC machinists convert blueprint fractional specifications into decimal coordinates to maintain tight imperial tolerance standards.

Common Pitfalls & Mistakes to Avoid

Key calculation errors to avoid when computing integration by parts calculator:

Adding Denominators Directly When Adding Fractions

Never add denominators (e.g. 1/2 + 1/3 ≠ 2/5). Find a common denominator first: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

Forgetting to Invert the Divisor When Dividing Fractions

Division by a fraction is equivalent to multiplication by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c). Keep, Change, Flip.

Canceling Terms in Fractions Instead of Common Factors

You can only cancel common multiplicative factors across numerator and denominator, not individual additive terms (e.g. (x + 2)/2 does not equal x + 1).

Key Terminology Glossary

Essential terms and definitions related to integration by parts calculator:

u (differentiable function) The u (differentiable function) input parameter for the Integration by Parts Calculator. Enter numerical values to execute calculations.
dv (integrable function) The dv (integrable function) input parameter for the Integration by Parts Calculator. Enter numerical values to execute calculations.
Numerator & Denominator In a fraction a/b, numerator a represents parts taken; denominator b represents total equal parts.
Greatest Common Divisor (GCD) The largest integer that divides both numerator and denominator without remainder, used to simplify fractions.
Verified STEM Methodology

About the Integration by Parts Calculator

The Integration by Parts Calculator is maintained by Basic Math Tools, an educational platform committed to providing accurate STEM and financial computing tools. Every tool processes calculations transparently in your browser for privacy, instant responsiveness, and mathematical accuracy.

If you have suggestions or questions regarding mathematical formulas, please review our Editorial Policy or contact our math team.

Fact-Checked & Verified • Computational Accuracy Standards
Updated August 2026 • Editorial Policy
Authored By
Sanjay Samanta

Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.

Reviewed & Verified By
Academic Review Board

Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.

Found an error or have an improvement suggestion? Report a calculation issue

Frequently Asked Questions

Does the Integration by Parts Calculator automatically reduce fractions to lowest terms?
Yes. The calculator divides both the numerator and denominator by their Greatest Common Factor (GCF) to express the final result in irreducible simplest form (e.g., 8/12 simplifies to 2/3).
Why must fractions have a common denominator before adding or subtracting?
Fractions can only be combined when their fractional parts represent equal division sizes. You cannot directly add 1/3 and 1/4 because thirds and fourths are different partition sizes. Finding the Least Common Denominator (LCD = 12) converts them to equal sizes (4/12 + 3/12 = 7/12) so numerators can be added directly.
Why do you invert and multiply when dividing fractions?
Dividing by a number is mathematically identical to multiplying by its multiplicative inverse (reciprocal). Asking "what is 1/2 divided by 1/4?" asks how many quarters fit inside a half: (1/2) ÷ (1/4) = (1/2) × (4/1) = 2.
How do I convert between improper fractions and mixed numbers?
To convert an improper fraction (e.g., 17/5) to a mixed number, divide the numerator by the denominator: 17 ÷ 5 = 3 with a remainder of 2, giving 3 2/5. To convert back, multiply whole number by denominator and add numerator: (3 × 5) + 2 = 17, giving 17/5.
How are decimal numbers converted into exact fractions?
Place the decimal digits over the corresponding power of 10 based on place value (e.g., 0.375 has 3 decimal places, so write 375/1000). Then divide both numerator and denominator by their greatest common factor (GCF = 125) to reduce to 3/8. For repeating decimals, algebraic equations (10x - x) eliminate the repeating tail.

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