INTEGRALS CONTAINING QUADRATIC POLYNOMIALS (SPECIAL CASES)

[Last Name] 1


INTEGRATION

INTEGRALS CONTAINING QUADRATIC POLYNOMIALS (SPECIAL CASES)

CASE 1

𝟏
Many Integrals in these general forms, ∫ (𝒂𝒙𝟐+𝒃𝒙+𝒄) 𝒅𝒙,

𝟏
∫( ) 𝒅𝒙, 𝒄𝒂𝒏 𝒃𝒆 𝒔𝒊𝒎𝒑𝒍𝒊𝒇𝒊𝒆𝒅 𝒃𝒚 𝒕𝒉𝒆 𝒑𝒓𝒐𝒄𝒆𝒔𝒔 𝒐𝒇 𝒄𝒐𝒎𝒑𝒍𝒆𝒕𝒊𝒏𝒈 𝒕𝒉𝒆 𝒔𝒒𝒖𝒂𝒓𝒆.
√𝒂𝒙𝟐 +𝒃𝒙+𝒄


This method is applied in order to transform such Integrals to their standard form, giving

room for the Appropriate integration technique to be easily used.

𝟏
Worked Example 1: Find ∫ (𝟗𝒙𝟐+𝟔𝒙+𝟓) 𝒅𝒙

Completing the square

9x²+6x+5

𝟔𝒙
𝟗 (𝒙𝟐 + )+𝟓
𝟗

𝟐𝒙
𝟗 (𝒙𝟐 + )+𝟓
𝟑


𝟏 𝟐 𝟏
𝟗 ((𝒙 + 𝟑) − 𝟗) + 𝟓

𝟏 𝟐
𝟗 (𝒙 + ) − 𝟏 + 𝟓
𝟑


𝟏 𝟐
𝟗 (𝒙 + 𝟑) + 𝟒

𝟏 𝟐
𝟑𝟐 (𝒙 + ) + 𝟒
𝟑

𝟐
𝟏
(𝟑 (𝒙 + 𝟑)) + 𝟒

(𝟑𝒙 + 𝟏)𝟐 + 𝟒

Re-writing the Integral

, [Last Name] 2

𝟏
∫ ((𝟑𝒙+𝟏)𝟐+𝟒) 𝒅𝒙

Applying the substitution, U=3x+1

𝒅𝒖
= 𝟑
𝒅𝒙

Making “dx” the subject

𝒅𝒖 𝟏
𝒅𝒙 = = 𝟑 𝒅𝒖
𝟑

Substituting the value of “dx” into the Integral and re-writing the Integral in terms of U

𝟏 𝟏
∫ (𝒖𝟐+𝟒) . 𝟑 𝒅𝒖

Pulling out ⅓

𝟏 𝟏
∫ (𝒖𝟐+𝟒) 𝒅𝒖
𝟑


Now, integrating by trigonometric substitution

𝟏 𝟏 𝒙
Recall, ∫ (𝒙𝟐+𝒂𝟐) 𝒅𝒙 = 𝒂 𝒕𝒂𝒏−𝟏 (𝒂) + 𝒄

Re-writing the Integral in the required pattern for the trigonometric substitution

𝟏 𝟏
∫ (𝒖𝟐+𝟐𝟐) 𝒅𝒖
𝟑


Where x=u, and a=2

We would then have;

𝟏 𝟏 𝒖
[ 𝒕𝒂𝒏−𝟏 ( 𝟐)]
𝟑 𝟐


Removing the bracket

𝟏 𝒖
𝒕𝒂𝒏−𝟏 ( 𝟐) + 𝒄
𝟔


Recall, U=3x+1

𝟏 𝟏 𝟑𝒙+𝟏
∫ (𝟗𝒙𝟐+𝟔𝒙+𝟓) 𝒅𝒙 = 𝟔 𝒕𝒂𝒏−𝟏 ( )+𝒄
𝟐