Integration, Completing the Square, Examples, Worksheet, Practice Problems - Calculus

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

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