The x-interval is transformed, scaled, integrated, and reconciledLive current inputs
Math & Statistics
Integral Step-by-Step Calculator
Work through a definite u-substitution for kx(ax^2+b)^p, including the differential scale factor, transformed bounds, power-rule antiderivative, and direct numerical reconciliation.
Exact definite integral -
Substitution scale k/(2a) -
Transformed lower bound -
Transformed upper bound -
Antiderivative power p+1 -
Direct numerical check -
Reconciliation residual -
Bound orientation -
U-SUBSTITUTION MACHINE
The x-interval is transformed, scaled, integrated, and reconciled
A four-stage diagram shows exactly where x dx becomes du/(2a), how the bounds move, and where the outside multiplier enters.
TRANSFORMATION LEDGER
Each algebraic object before and after substitution
The register prevents a frequent error: changing the integrand to u while leaving the original x-bounds in place.
| Object | In x-space | Transformation | In u-space | Current value |
|---|
DERIVATION SETUP
Confirm the derivative pattern before invoking substitution
- Identify the complete inner expression ax squared plus b.
- Confirm the integrand also contains x dx.
- Keep the multiplier k visible until the differential scale is formed.
- Transform both bounds immediately after defining u.
- Use the numerical residual only as a check on the derived exact expression.
PATTERN MATCH
Substitution is a change of variable, not a label swap
The success of the method depends on the outside factor matching the derivative of the inner expression up to a constant. Here, x dx supplies the needed linear term.
Definite integration is especially sensitive to mixed variables. Once the bounds are converted to u, the antiderivative must be evaluated entirely in u-space.
POWER SUBSTITUTION
Choose u so the remaining x dx is exactly proportional to du
With u=ax^2+b, du=2ax dx. The factor k/(2a) remains outside, and both definite bounds must be transformed before applying the power rule.
Detailed calculation process and general formulas
u=ax^2+bdu=2ax dxkx dx=(k/(2a))duIntegral=(k/(2a))*[u^(p+1)/(p+1)]_(u_l)^(u_u)u_l=a l^2+b; u_u=a u^2+bSymbols, meanings, and units
- u
- substituted inner quadraticinner-expression unit
- p
- power on the inner expressiondimensionless
- k
- outside multiplierintegrand-specific
- a
- quadratic coefficient inside uinner/x^2
- l,u
- entered x-boundsx-unit
- du
- differential of the substituted variableu-unit
ERROR-PROOFING
Three checkpoints that catch most substitution mistakes
The live machine exposes scale, bounds, and exponent as separate decisions.
Differential scale
-The outside multiplier becomes k divided by 2a.
Bound conversion
-Both x endpoints are mapped through the same inner expression.
Power increment
-The u exponent increases by one before division.
Decision takeaway: Do not integrate until the differential, bounds, and remaining integrand all use the same variable.
DERIVATION NOTES
Cases requiring a different method
- p=-1 produces a logarithm and is excluded by the input range.
- Negative inner values with fractional p can leave the real-number domain.
- An outside factor that is not proportional to x requires another approach.
- Reversed x-bounds preserve orientation through transformed endpoint evaluation.
Applied decisions
How substitution choices change the difficulty
Exact derivative match
The integrand contains x times a power of ax squared plus b.
What the result clarifies: Only a constant scale remains after substitution.
Missing x factor
The same powered quadratic appears without x dx.
What the result clarifies: The displayed pattern does not apply, so forcing u would leave x unresolved.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This page covers the stated power-substitution family with positive inner constants and p greater than -1. It does not choose a method for arbitrary integrals or handle complex-valued branches.
Integral Step-by-Step Calculator | Power Substitution Derivation FAQ
Why divide by 2a?
Because du=2ax dx, so x dx=du/(2a).
Why transform the bounds?
After substitution the antiderivative is a function of u, so its endpoints must also be u-values.
Where did the integration constant go?
It cancels in a definite endpoint difference.
What if the bounds have equal squares?
The transformed bounds match, and the definite integral of this odd-in-x structure over that orientation reconciles to zero.