Integration by completing the square is a method for evaluating integrals with quadratic expressions in the denominator, particularly when the quadratic is irreducible (cannot be factored into real linear terms). The process involves rewriting the quadratic \(ax^{2}+bx+c\) into the form \(a(x+k_{1})^{2}+k_{2}\), which is then manipulated using u-substitution into a standard integral form, typically involving the inverse tangent (\(\arctan \)) or inverse sine (\(\arcsin \)) functions. 💡Steps for Completing the Square • Ensure x² coefficient is one: If the coefficient of \(x^{2}\) (let's call it \(a\)) is not 1, factor it out from the entire quadratic expression. ∘ For example, \(2x^{2}-4x+11\) becomes \(2(x^{2}-2x+\frac{11}{2})\). • Complete the square on the remaining quadratic: Take the coefficient of the \(x\) term (let's call it \(b^{\prime }\)), divide it by 2, and then square the result. Add and subtract this value inside the parenthesis. ∘ For example, in \(x^{2}-2x+\frac{11}{2}\), the \(x\) coefficient is -2. ∘ Half of -2 is -1, and squaring it gives 1. ∘ The expression becomes \(2(x^{2}-2x+1+\frac{11}{2}-1)\). • Factor the perfect square trinomial: The first three terms inside the parenthesis now form a perfect square trinomial, \((x+b^{\prime }/2)^{2}\). ∘ \(2((x-1)^{2}+\frac{9}{2})\). • Simplify the constant term: Combine the remaining constant terms. ∘ \(2((x-1)^{2}+\frac{9}{2})=2(x-1)^{2}+9\). 💡Applying to Integrals • Rewrite the denominator: Substitute the completed square form into the integral. ∘ For example, \(\int \frac{1}{2x^{2}-4x+11}dx\) becomes \(\int \frac{1}{2(x-1)^{2}+9}dx\). • Use substitution: Let \(u\) be the term in the parenthesis. In the example, \(u=x-1\), so \(du=dx\). ∘ The integral becomes \(\int \frac{1}{2u^{2}+9}du\). • Manipulate to match standard forms: Factor out constants and rearrange the expression to match a known integral, often involving the form \(\frac{1}{a^{2}+u^{2}}\). ∘ \(\int \frac{1}{2u^{2}+9}du=\frac{1}{2}\int \frac{1}{u^{2}+(3/\sqrt{2})^{2}}du\). • Integrate: Apply the appropriate integral formula. ∘ The integral \(\int \frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}\arctan (\frac{x}{a})+C\). ∘ The final result can be converted back to the original variable. 💡When to Use This Method • Irreducible quadratics: This method is essential for integrals involving quadratic denominators that do not factor into real linear terms. • Non-standard forms: It is useful for integrands that resemble natural logarithms or inverse trigonometric functions but are "off" due to the quadratic expression. • Standard forms: Completing the square transforms a general quadratic into a form that can be solved using known integration techniques, such as those found in integral tables. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Completing the square 00:40 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng