C51 · Unit C6 Integration and accumulation
Completing the square inside an integral
CoreReview editionNot yet human-reviewed
- Time
- About 45 minutes, untimed
- Course framework reference
- AP 6.10
- Builds on
- Part of
Progress on this lesson counts in every pathway listed.
Understand the idea
A quadratic denominator may become an inverse-tangent integral after completing the square. Rewrite x squared +bx+c equals (x+b over 2) squared +c minus b squared /4. If the final constant is positive, a shift and scale converts it to 1/(1+u squared). If it is zero or negative, a different method and domain analysis are needed.
Worked example
∫dx/(x squared +4x+8) equals ∫dx/[(x+2) squared +4]. Set u equals (x+2)/2, dx equals 2du. The result is one half arctan((x+2)/2)+C. Differentiation confirms both the shift and the factor.
Explore the mathematics
Completing the square inside an integral
Rewrite a quadratic denominator to reveal an inverse-trigonometric integral.
Scroll sideways to see the full diagram.
Predict before continuing
Can completing the square reveal a familiar antiderivative?
Read the complete explanation
- Can completing the square reveal a familiar antiderivative?
- x²+2x+2=(x+1)²+1.
- Use arctan(x+1); its derivative is 1/[1+(x+1)²].
- The definite value is arctan(2)−π/4.
∫dx/(x²+4x+8)=∫dx/[(x+2)²+4]. Set u=(x+2)/2, dx=2du. The result is ½arctan((x+2)/2)+C. Differentiation confirms both the shift and the factor.
Discuss: explain what each label represents, then draw a related example with different values.
A common misconception
Completing the square changes the constant: x squared +4x+8 is (x+2) squared +4, not (x+2) squared +8.
Investigate and explain
Compare quadratic denominators with positive, zero and negative completed constants. Predict which antiderivative family is appropriate.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.