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Integration methods

U-Substitution vs. Integration by Parts: How to Choose

The fastest way to choose an integration method is to inspect structure before manipulating symbols. A visible product does not automatically mean integration by parts, and an inner expression does not help unless its derivative is also present up to a constant factor.

Direct answer

Use substitution for a composite pattern f(g(x))g′(x); use integration by parts when a product becomes simpler after differentiating one factor.

This is a method-selection test, not a guarantee that either technique finishes every integral.

Decision method: inspect what becomes simpler

Start with substitution. Look for an inner expression g(x) inside a power, exponential, logarithm or trigonometric function. If the remaining factor is g′(x), possibly multiplied by a constant, set u=g(x). For ∫2x cos(x²) dx, u=x² and du=2x dx reduce the integral to ∫cos(u) du.

If the integrand is a product and substitution does not absorb one factor, test integration by parts. Choose u so that differentiating it makes the expression simpler, and choose dv so that it can be integrated. For ∫x exp(x) dx, taking u=x and dv=exp(x)dx gives exp(x)(x−1)+C.

Do not decide from appearance alone. The product x sin(x²) contains the derivative of x² up to a constant, so substitution works even though two factors are visible. By contrast, x sin(x) has no matching inner derivative; integration by parts exchanges it for the simpler integral ∫cos(x)dx.

After choosing a method, differentiate the proposed antiderivative. That check is more reliable than deciding that an answer looks familiar. CalculusMate currently handles a bounded deterministic set of antiderivatives; a mathematically valid method may still be reported as unsupported when that rule is not implemented.

1
Composite + matching inner derivative → substitution
2
Product + simpler differentiated factor → integration by parts
3
Differentiate the result to check it

Worked examples

∫ 2x cos(x²) dx
sin⁡(x2)+C\sin\left(x^{2}\right) + C

Use u=x² because du=2x dx appears in the integrand.

∫ x exp(x) dx
ex (x−1)+Ce^{x}\,\left(x - 1\right) + C

Use parts with u=x and dv=exp(x)dx; differentiating x removes the polynomial factor.

∫e2 x dx\int e^{2\,x}\,\mathrm{d}x
e2 x2+C\frac{e^{2\,x}}{2} + C

This is a linear-inner substitution pattern supported by the current deterministic calculator.

Common mistakes

  • • Choosing integration by parts merely because the integrand is written as a product.
  • • Choosing u=g(x) when no multiple of g′(x) remains in the integrand.
  • • Stopping after substitution without converting du correctly or returning to x.
  • • Assuming an unsupported calculator result means the mathematical method is invalid.

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

The linked exp(2x) example is supported. The two broader method-selection examples above are educational and may remain outside the current deterministic rule set.

积分:e2 xe^{2\,x}
Open the calculator without a preset: 积分

References

Frequently asked questions

Can a product still use u-substitution?+

Yes. If one factor is the derivative of an inner expression in the other factor, substitution may be the shorter method.

How should I choose u in integration by parts?+

Prefer a factor that becomes simpler when differentiated, while the remaining factor can be integrated without making the problem harder.

What if both methods make the integral harder?+

Recheck algebra and consider another technique, such as an identity, partial fractions or a numerical method. Not every integral has an elementary antiderivative.

Conclusion

Choose from structure: a matching inner derivative points to substitution, while a product with a simplifiable factor points to integration by parts. Then differentiate the result and keep calculator limitations separate from the mathematics.

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