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Integrals of Some Particular Functions - III

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Integrals of Some Particular Functions - III - Lesson Summary

These integrals follow some definite pattern, and solved in a particular way.

∫ (px + q)/(ax2 + bx + c) dx   (or) ∫ (px + q)/√(ax2 + bx + c) dx


1) I = ∫ (px + q)/(ax2 + bx + c) dx

px+q = A d/dx(ax2 + bx + c) + B

px+q = A(2ax + b) + B

px+q = 2Aax + Ab + B

Comparing the coefficients of x:

p = 2Aa  ⇒ A = p/2a

Comparing the constants:

q = Ab + B ⇒ B = q - Ab

px+q = p/2a . (2ax + b) + (q - Ab)

I = ∫ (p/2a (2ax+b) + (q-Ab))/(ax2 + bx + c) dx

I = p/2a ∫ (2ax+b)/(ax2 + bx + c) dx + ∫ (q-Ab)/(ax2 + bx + c) dx

I = I1 + I2

I1 = p/2a ∫ (2ax+b)/(ax2 + bx + c) dx

I2 = ∫ (q-Ab)/(ax2 + bx + c) dx = (q-Ab) ∫ 1/(ax2 + bx + c) dx

ax2 + bx + c = t ⇒ (2ax + b)dx = dt

⇒ I1 = p/2a ∫ dt/t = p/2a log|t| + C1

⇒ I1 = p/2a log|ax2 + bx + c| + C1

I2 = (q-Ab) ∫ 1/(ax2 + bx + c) dx = (q-Ab)/a ∫ 1/((x+b/2a)2 + (√(4ac-b2)/4a2)2 dx 

I = p/2a log(ax2 + bx + c) + (q-Ab)/2.√(b - a2/4) log |((x + a/2) - √(b - a2/4))/((x + a/2) + √(b - a2/4))| + C

(or)

I = p/2a log(ax2 + bx + c) + (q-Ab)/2.√(b - a2/4) log |(√(b - a2/4) - (x + a/2))/(√(b - a2/4) - (x + a/2))| + C

(or)

I = p/2a log(ax2 + bx + c) + 1/√(b - a2/4) . tan-1(x + a/2)/√(b - a2/4) + C


2) ∫ (px+q)/√(ax2+bx+c) dx

px + q = A d/dx(ax2 + bx + c) + B

px + q = A(2ax + b) + B

px + q = 2Aax + Ab + B

Comparing the coefficients of x:

px = 2Aax  ⇒ p/2a

Comparing the constants:

q = Ab + B ⇒ B = q - Ab

px + q = p/2a . (2ax + b) + (q - Ab)

I = ∫ (p/2a (2ax + b)+ (q - Ab))/√(ax2 + bx + c) dx

I = p/2a ∫ 2ax+b / √(ax2 + bx + c) dx + (q - Ab) ∫ 1/√(ax2 + bx + c) dx

I = I1 + I2

I1 = p/2a ∫ 2ax+b / √(ax2 + bx + c) dx

ax2 + bx + c = t   ⇒ (2ax + b)dx = dt

⇒ I1 = p/2a ∫ dt/√t

= p/2a 2√t + C1 = p/a . √t + c1

⇒ I1 = p/a . (√(ax2 + bx + c)) = p/a .(√(ax2 + bx + c)) + c1 

I2 = (q - Ab) ∫ 1/√(ax2 + bx + c) dx = (q - Ab) x log|(x+a/2) + √((x+a/2)2 - (√(b - a2/4))2)| + C
                                                    or
I2 = (q - Ab) ∫ 1/√(ax2 + bx + c) dx = (q - Ab) x sin-1( (x+a/2)/(√(b - a2/4)) ) + C
                                                    or
I2 = (q - Ab) ∫ 1/√(ax2 + bx + c) dx = (q - Ab) x log|(x+a/2) + √((x+a/2)2 + (√(b - a2/4))2)| + C

= (q - Ab) ∫ 1/√((x+a/2)2 + (√(b-a2/4)2))

I2 = p/a . (√(ax2 + bx + c)) + (q - Ab) x log|(x+a/2) + √((x+a/2)2 + (√(b - a2/4))2)| + C

(or)

I2 = p/a .(√(ax2 + bx + c)) + (q - Ab) x sin-1( (x+a/2)/(√(b - a2/4)) ) + C

(or)

I2 = p/a . (1/√(ax2 + bx + c)) + log|(x+a/2) + √((x+a/2)2 + (√(b - a2/4))2)| + C

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