Calculator Input
Example Data Table
| Case | u | dv/dx | v | Common Result Form |
|---|---|---|---|---|
| Polynomial with trig | x | cos(x) | sin(x) | x sin(x) - ∫sin(x) dx |
| Polynomial with exponential | x^2 | exp(x) | exp(x) | x^2 exp(x) - ∫2x exp(x) dx |
| Logarithmic | ln(x) | 1 | x | x ln(x) - ∫1 dx |
| Scaled trig | x | sin(2*x) | -0.5 cos(2*x) | -0.5x cos(2x) + 0.5∫cos(2x) dx |
Formula Used
The calculator uses the standard integration by parts identity:
∫u dv = uv - ∫v du
You enter u and the expression for dv/dx. The tool finds a basic derivative for u and a basic antiderivative for dv. It then places those values into the identity.
For definite bounds, it also checks the value numerically. Simpson estimation compares the direct integral against the right side of the parts formula.
How to Use This Calculator
- Enter the variable used in the integral.
- Type the expression selected as u.
- Type the dv/dx expression, such as
cos(x). - Add lower and upper bounds only when you need a definite estimate.
- Choose Simpson intervals for numerical checking.
- Press the calculate button.
- Review the result shown above the form.
- Download the result as CSV or PDF when needed.
Article: Understanding Integration by Parts
What the Method Does
Integration by parts is a key rule in calculus. It helps when a product is hard to integrate directly. The method rewrites the product into a simpler form. One factor is chosen as u. The other factor becomes dv. Good choices make the next integral easier. Poor choices can make the work longer.
Choosing u and dv
A common guide is LIATE. It ranks logarithmic terms first. Then come inverse trig terms. Algebraic terms are usually next. Trig and exponential terms often work well as dv. This rule is not perfect. It is still a helpful starting point. This calculator gives a short choice hint.
Why Steps Matter
The parts formula has several moving pieces. You need u, dv, du, and v. A small sign error can change the final answer. Step output helps you inspect each piece. It also helps students compare class work. The formula line keeps the structure visible.
Definite Integral Checking
Definite bounds add another layer. The calculator estimates the original product integral. It also estimates the integration by parts right side. The difference shows how close both paths are. Smaller differences usually mean better numerical agreement. More Simpson intervals can improve smooth examples.
Practical Use
Use this page for homework checks. Use it to prepare examples. Use exports to save work records. Enter expressions with explicit multiplication. Review the symbolic result before relying on it. Some advanced expressions need manual simplification. The calculator is strongest for common teaching examples.
FAQs
1. What is integration by parts?
It is a calculus method for integrating products. It rewrites ∫u dv as uv - ∫v du. The goal is to make the remaining integral easier than the original one.
2. How should I choose u?
Use LIATE as a guide. Prefer logarithmic terms first, then inverse trig, algebraic, trig, and exponential terms. The best choice should simplify after differentiation.
3. What should I enter for dv?
Enter the expression that multiplies u inside the integral. For ∫x cos(x) dx, choose u as x and enter cos(x) as dv/dx.
4. Can this solve every integral?
No. It handles many common symbolic patterns. More complex expressions may need manual simplification or a dedicated algebra system after the parts setup.
5. Why use Simpson checking?
Simpson checking estimates a definite integral numerically. It helps compare the original integral with the integration by parts right side when bounds are provided.
6. Why did I get “not available”?
The expression may use unsupported syntax or a function outside the simple evaluator. Add multiplication signs and use common functions like sin, cos, exp, ln, and sqrt.
7. Does the calculator include the constant?
Yes. For indefinite forms, the selected constant symbol is added to the step result. Definite estimates do not need a constant.
8. Can I export my answer?
Yes. After calculating, use the CSV or PDF buttons above the form. They download the latest result stored during the current session.