Algebraic times exponential
- Input
- ∫ x·e^x dx
- Expected output
- x·eˣ − eˣ + C
u = x, dv = eˣ dx.
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integration by parts calculator
Integration by parts shows up when the integrand is a product of functions from different families, such as x·eˣ, x·ln x or x²·sin x. The calculator picks u by the LIATE order, computes du and v, builds u·v − ∫ v du and shows each round, including when the method has to be repeated.
u = x, dv = eˣ dx.
u = ln x, dv = dx.
Each round lowers the degree of x.
The original integral reappears and is isolated.
u = ln x, v = x²/2.
The indefinite integral of f is the family of functions F whose derivative is f. Since the derivative of a constant is zero, if F is an antiderivative then F + C is too, for any number C. That is why ∫ 2x dx = x² + C: x² + 1 and x² − 7 also have derivative 2x. The constant can only be pinned down with an extra condition, such as F(0) = 3.
When ∫ v du becomes harder than the original. Then swap u and dv. If the integrand is composite, as in 2x·cos(x²), the substitution u = x² solves it in one step and is the right method. The calculator tries the shorter rules first and only resorts to parts when they fail.
It uses LIATE and, if the first choice of u does not close, tries the second one on the list. The two variants only differ in the position of logarithmic and inverse trigonometric, which rarely appear together in the same product.
The check panel differentiates the result and compares it with the integrand at 14 test points. If the relative difference is below 1e-7 everywhere, the result is accepted; otherwise it is discarded.
Differentiating the result
F′(x) = f(x) at 14 test points (maximum error 2.2e-16). Both expressions are also identical after expanding.
| x | f(x) | F′(x) | Difference |
|---|---|---|---|
| -2.26376 | -0.23533751 | -0.23533751 | 2.8e-17 |
| 0.565523 | 0.99552794 | 0.99552794 | 1.1e-16 |
| 4.44149 | 377.08727 | 377.08727 | 5.7e-14 |
| -0.222645 | -0.17820488 | -0.17820488 | 0 |
| -0.357476 | -0.25003303 | -0.25003303 | 0 |
| 0.460526 | 0.72989202 | 0.72989202 | 2.2e-16 |
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