HOW TO INTEGRATE QUADRATIC EQUATION IN THE SQUARE ROOT

To know the formulas used in integration, please visit the page "Integration Formulas for Class 12".

Question 1 :

Evaluate the following with respect to "x".

 √(x² + 2x + 10)

Solution :

 ∫ √(x² + 2x + 10) dx 

x² + 2x + 10  =  x² + 2x(1) + 1² - 1² + 10 

=  (x + 1)² - 1 + 10 

=  (x + 1)² + 9 

=  (x + 1)² + 3²

 ∫ √(x² + 2x + 10) dx  =   ∫√[(x + 1)² + 3²] dx 

=  ((x+1)/2)√(x²+2x+10) + (9/2)log (x+√(x²+2x+10)) + c

Question 2 :

Evaluate the following with respect to "x".

 √(x² - 2x - 3)

Solution :

 √(x² - 2x - 3) dx 

x² - 2x - 3  =  x² - 2x(1) + 1² - 1² -3

=  (x - 1)² - 1 - 3 

=  (x - 1)² - 4 

=  (x - 1)² - 2²

 ∫  √(x² - 2x - 3) dx  =   ∫√[(x - 1)² - 2²] dx 

=  ((x-1)/2)√(x² - 2x - 3) - (4/2)log (x-1+√(x²-2x-3)) + c

=  ((x-1)/2)√(x² - 2x - 3) - 2 log (x-1 + √(x² - 2x - 3)) + c

Question 3 :

Evaluate the following with respect to "x".

 √(6 - x)(x - 4)

Solution :

 √(6 - x)(x - 4) dx 

(6 - x)(x - 4)  =  6x - 24 - x² + 4x

  =  -x² + 10x - 24

  =  -[x² - 10x + 24]

=  -[x² - 2x(5) + 5² - 5² + 24]

=  -[(x-5)² - 25 + 24]

=  -[(x-5)² - 1]

=  1² - (x-5)²

∫(√a²-x²) dx = (x/2)(√a²-x²)+(a²/2) sin^-1(x/a) + c

 ∫√(-x² + 10x - 24) dx  =   ∫√[1² + (x-5)²] dx 

=  ((x-5)/2)√(-x² + 10x - 24)+(1/2)sin^-1((x-5)/1) + c

Related topics

• Indefinite Integral Using Properties
• Integration of Rational Functions with Quadratic Denominator
• Integration of Rational Functions With Square Roots
• Integration of Linear in Numerator and Quadratic in the Denominator
• How to Integrate a Rational Function Linear in the Numerator
• Integration of Rational Function With Square Root in Denominator

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