INDEFINITE INTEGRALS WITH SQUARE ROOTS

About "How to Integrate Quadratic Equation in the Square Root"

How to Integrate Quadratic Equation in the Square Root :

Here we are going to see some example problems to understand integration quadratic equations in the  square root.

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

Integration of Quadratic Equation in the Square Root - Examples

Question 1 :

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

Question 2 :

Evaluate the following with respect to "x".

 √[9 - (2x + 5)²]

Solution :

 ∫√[9 - (2x + 5)²] dx 

√[9 - (2x + 5)²] =  √[3² - (2x + 5)²] dx

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

=  (2x + 5)/2√[9 - (2x + 5)²] + (9/2) sin^-1[(2x + 5)/3] + c

Question 3 :

Evaluate the following with respect to "x".

 √[81 + (2x + 1)²]

Solution :

 ∫√[81 + (2x + 1)²] dx 

√[81 + (2x + 1)²]  =  √[9² + (2x + 1)²] dx

∫(√x²+a²) dx = (x/2)(√x²+a²)+(a²/2) log (x+√(x²+a²)+c

=  ((2x+1)/3)√[9²+(2x+1)²]+(9/2) log (2x+1)+√[9²+ (2x+1)²] 

=  ((2x+1)/3)√[81 + (2x + 1)²]+(9/2) log (2x+1)+√[81 + (2x + 1)²] + c

Question 4 :

Evaluate the following with respect to "x".

 √[(x + 1)² - 4]

Solution :

 ∫ √[(x + 1)² - 4] dx

 ∫ √[(x + 1)² - 4]  =  √[(x - 1)² - 2²] dx

∫(√x²-a²)dx=(x/2)(√a²-x²)-(a²/2)log(x+ √a²-x²) + c

=  ((x-1)/2)√[(x + 1)² - 4] - (4/2) log (x - 1 - √[(x + 1)² - 4] + 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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