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

After having gone through the stuff given above, we hope that the students would have understood, "Indefinite Integrals With Square Roots"

Apart from the stuff given in "Indefinite Integrals With Square Roots", if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.