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CalculusBy Educator Today

C25 · Unit C6 Integration and accumulation

Substitution and algebra before integration

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 6.9–6.10
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

Substitution reverses the chain rule. Choose u equals g(x), replace g minutes (x)dx by du, and rewrite the entire integrand. In a definite integral either change both bounds to u-values or return fully to x before using the original bounds. Algebraic division and completing the square can expose a standard antiderivative before substitution.

Worked example

∫ sub 0 to the power 1 2x(x squared +1) cubed dx: let u equals x squared +1, du equals 2x dx, with bounds 1 to 2. The result is [u to the power 4 /4] sub 1 squared equals 15 over 4. Separately, (x squared +1)/(x+1) equals x minus 1+2/(x+1), so polynomial division simplifies its integral.

Explore the mathematics

Substitution and algebra before integration

Change a variable and its differential and limits together.

Setting 3 of 5

Scroll sideways to see the full diagram.

Look for an inside expression and its derivative.xy1Marked input 0.5; displayed curve height 1.25∫₀¹ 2x(x²+1) dx

Predict before continuing

Look for an inside expression and its derivative.

Read the complete explanation
  1. Look for an inside expression and its derivative.
  2. Set u=x²+1, so du=2x dx. The limits become u=1 and u=2.
  3. Integrate ∫₁² u du=[u²/2]₁².
  4. The result is 3/2. If you change variables, transform the limits or substitute back before using the original limits.
Substitution and algebra before integration · visual worked method
Substitution and algebra before integration · visual worked method∫₀¹2x(x²+1)³dx: let u=x²+1, du=2x dx, with bounds 1 to 2. The result is [u⁴/4]₁²=15/4. Separately, (x²+1)/(x+1)=x−1+2/(x+1), so polynomial division simplifies its integral.Worked example · 1/2∫₀¹2x(x²+1)³dx: let u=x²+1, du=2xdx, with bounds 1 to 2. The resultis [u⁴/4]₁²=15/4. Separately,(x²+1)/(x+1)=x−1+2/(x+1),Worked example · 2/2so polynomial division simplifiesits integral.

∫₀¹2x(x²+1)³dx: let u=x²+1, du=2x dx, with bounds 1 to 2. The result is [u⁴/4]₁²=15/4. Separately, (x²+1)/(x+1)=x−1+2/(x+1), so polynomial division simplifies its integral.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Do not mix x-bounds with a u-antiderivative. A substitution must account for every remaining factor.

Investigate and explain

Sort integrals into direct rule, algebra first, and substitution. Explain your proposed first move without completing every calculation.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.