Notes On Integrals of Some Particular Functions - III - CBSE Class 12 Maths
These integrals follow some definite pattern, and solved in a particular way. âˆ« (px + q)/(ax2 + bx + c) dx   (or) âˆ« (px + q)/âˆš(ax2 + bx + c) dx 1) I = âˆ« (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  â‡’ A = p/2a Comparing the constants: q = Ab + B â‡’ B = q - Ab px+q = p/2a . (2ax + b) + (q - Ab) I = âˆ« (p/2a (2ax+b) + (q-Ab))/(ax2 + bx + c) dx I = p/2a âˆ« (2ax+b)/(ax2 + bx + c) dx + âˆ« (q-Ab)/(ax2 + bx + c) dx I = I1 + I2 I1 = p/2a âˆ« (2ax+b)/(ax2 + bx + c) dx I2 = âˆ« (q-Ab)/(ax2 + bx + c) dx = (q-Ab) âˆ« 1/(ax2 + bx + c) dx ax2 + bx + c = t â‡’ (2ax + b)dx = dt â‡’ I1 = p/2a âˆ« dt/t = p/2a log|t| + C1 â‡’ I1 = p/2a log|ax2 + bx + c| + C1 I2 = (q-Ab) âˆ« 1/(ax2 + bx + c) dx = (q-Ab)/a âˆ« 1/((x+b/2a)2 + (âˆš(4ac-b2)/4a2)2 dx  I = p/2a log(ax2 + bx + c) + (q-Ab)/2.âˆš(b - a2/4) log |((x + a/2) - âˆš(b - a2/4))/((x + a/2) + âˆš(b - a2/4))| + C (or) I = p/2a log(ax2 + bx + c) + (q-Ab)/2.âˆš(b - a2/4) log |(âˆš(b - a2/4) - (x + a/2))/(âˆš(b - a2/4) - (x + a/2))| + C (or) I = p/2a log(ax2 + bx + c) + 1/âˆš(b - a2/4) . tan-1(x + a/2)/âˆš(b - a2/4) + C 2) âˆ« (px+q)/âˆš(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  â‡’ p/2a Comparing the constants: q = Ab + B â‡’ B = q - Ab px + q = p/2a . (2ax + b) + (q - Ab) I = âˆ« (p/2a (2ax + b)+ (q - Ab))/âˆš(ax2 + bx + c) dx I = p/2a âˆ« 2ax+b / âˆš(ax2 + bx + c) dx + (q - Ab) âˆ« 1/âˆš(ax2 + bx + c) dx I = I1 + I2 I1 = p/2a âˆ« 2ax+b / âˆš(ax2 + bx + c) dx ax2 + bx + c = t   â‡’ (2ax + b)dx = dt â‡’ I1 = p/2a âˆ« dt/âˆšt = p/2a 2âˆšt + C1 = p/a . âˆšt + c1 â‡’ I1 = p/a . (âˆš(ax2 + bx + c)) = p/a .(âˆš(ax2 + bx + c)) + c1  I2 = (q - Ab) âˆ« 1/âˆš(ax2 + bx + c) dx = (q - Ab) x log|(x+a/2) + âˆš((x+a/2)2 - (âˆš(b - a2/4))2)| + C                                                     or I2 = (q - Ab) âˆ« 1/âˆš(ax2 + bx + c) dx = (q - Ab) x sin-1( (x+a/2)/(âˆš(b - a2/4)) ) + C                                                     or I2 = (q - Ab) âˆ« 1/âˆš(ax2 + bx + c) dx = (q - Ab) x log|(x+a/2) + âˆš((x+a/2)2 + (âˆš(b - a2/4))2)| + C = (q - Ab) âˆ« 1/âˆš((x+a/2)2 + (âˆš(b-a2/4)2)) I2 = p/a . (âˆš(ax2 + bx + c)) + (q - Ab) x log|(x+a/2) + âˆš((x+a/2)2 + (âˆš(b - a2/4))2)| + C (or) I2 = p/a .(âˆš(ax2 + bx + c)) + (q - Ab) x sin-1( (x+a/2)/(âˆš(b - a2/4)) ) + C (or) I2 = p/a . (1/âˆš(ax2 + bx + c)) + log|(x+a/2) + âˆš((x+a/2)2 + (âˆš(b - a2/4))2)| + C

#### Summary

These integrals follow some definite pattern, and solved in a particular way. âˆ« (px + q)/(ax2 + bx + c) dx   (or) âˆ« (px + q)/âˆš(ax2 + bx + c) dx 1) I = âˆ« (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  â‡’ A = p/2a Comparing the constants: q = Ab + B â‡’ B = q - Ab px+q = p/2a . (2ax + b) + (q - Ab) I = âˆ« (p/2a (2ax+b) + (q-Ab))/(ax2 + bx + c) dx I = p/2a âˆ« (2ax+b)/(ax2 + bx + c) dx + âˆ« (q-Ab)/(ax2 + bx + c) dx I = I1 + I2 I1 = p/2a âˆ« (2ax+b)/(ax2 + bx + c) dx I2 = âˆ« (q-Ab)/(ax2 + bx + c) dx = (q-Ab) âˆ« 1/(ax2 + bx + c) dx ax2 + bx + c = t â‡’ (2ax + b)dx = dt â‡’ I1 = p/2a âˆ« dt/t = p/2a log|t| + C1 â‡’ I1 = p/2a log|ax2 + bx + c| + C1 I2 = (q-Ab) âˆ« 1/(ax2 + bx + c) dx = (q-Ab)/a âˆ« 1/((x+b/2a)2 + (âˆš(4ac-b2)/4a2)2 dx  I = p/2a log(ax2 + bx + c) + (q-Ab)/2.âˆš(b - a2/4) log |((x + a/2) - âˆš(b - a2/4))/((x + a/2) + âˆš(b - a2/4))| + C (or) I = p/2a log(ax2 + bx + c) + (q-Ab)/2.âˆš(b - a2/4) log |(âˆš(b - a2/4) - (x + a/2))/(âˆš(b - a2/4) - (x + a/2))| + C (or) I = p/2a log(ax2 + bx + c) + 1/âˆš(b - a2/4) . tan-1(x + a/2)/âˆš(b - a2/4) + C 2) âˆ« (px+q)/âˆš(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  â‡’ p/2a Comparing the constants: q = Ab + B â‡’ B = q - Ab px + q = p/2a . (2ax + b) + (q - Ab) I = âˆ« (p/2a (2ax + b)+ (q - Ab))/âˆš(ax2 + bx + c) dx I = p/2a âˆ« 2ax+b / âˆš(ax2 + bx + c) dx + (q - Ab) âˆ« 1/âˆš(ax2 + bx + c) dx I = I1 + I2 I1 = p/2a âˆ« 2ax+b / âˆš(ax2 + bx + c) dx ax2 + bx + c = t   â‡’ (2ax + b)dx = dt â‡’ I1 = p/2a âˆ« dt/âˆšt = p/2a 2âˆšt + C1 = p/a . âˆšt + c1 â‡’ I1 = p/a . (âˆš(ax2 + bx + c)) = p/a .(âˆš(ax2 + bx + c)) + c1  I2 = (q - Ab) âˆ« 1/âˆš(ax2 + bx + c) dx = (q - Ab) x log|(x+a/2) + âˆš((x+a/2)2 - (âˆš(b - a2/4))2)| + C                                                     or I2 = (q - Ab) âˆ« 1/âˆš(ax2 + bx + c) dx = (q - Ab) x sin-1( (x+a/2)/(âˆš(b - a2/4)) ) + C                                                     or I2 = (q - Ab) âˆ« 1/âˆš(ax2 + bx + c) dx = (q - Ab) x log|(x+a/2) + âˆš((x+a/2)2 + (âˆš(b - a2/4))2)| + C = (q - Ab) âˆ« 1/âˆš((x+a/2)2 + (âˆš(b-a2/4)2)) I2 = p/a . (âˆš(ax2 + bx + c)) + (q - Ab) x log|(x+a/2) + âˆš((x+a/2)2 + (âˆš(b - a2/4))2)| + C (or) I2 = p/a .(âˆš(ax2 + bx + c)) + (q - Ab) x sin-1( (x+a/2)/(âˆš(b - a2/4)) ) + C (or) I2 = p/a . (1/âˆš(ax2 + bx + c)) + log|(x+a/2) + âˆš((x+a/2)2 + (âˆš(b - a2/4))2)| + C

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