INDEFINITE INTEGRALS WITH SQUARE ROOTS

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

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

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

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)- 4] dx

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

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