Notes On Integrals of some Particular Functions - I - CBSE Class 12 Maths
Formulae of integrals: âˆ«dx/(x2 - a2) = 1/2a . log |(x - a)/(x + a)| + C âˆ«dx/(a2 - x2) = 1/2a . log |(a - x)/(a + x)| + C âˆ«dx/(x2 + a2) = 1/a . tan-1 x/a + C 1) âˆ«1/(x2 - a2) dx = âˆ«1/((x + a)(x - a)) dx âˆ«1/((x + a)(x - a)) dx = 1/2a . âˆ« [1/(x-a) - 1/(x+a)] dx = 1/2a . âˆ« [1/(x-a)  dx- 1/(x+a) dx] = 1/2a . [log |x-a| - log |x+a|] + C = 1/2a . log|(x-a)/(x+a)| + C  ['.' log|a| - log|b| = log|a/b|] 2) âˆ« 1/(a2 - x2) dx âˆ« 1/((a+x)(a-x)) dx = 1/2a . âˆ«[1/(a+x) + 1/(a-x)] dx 1/2a [âˆ« 1/a+x dx + âˆ« 1/a-x dx] = 1/2a [log|a+x| - log|a-x|] + C = 1/2a log|(a+x)/(a-x)| + C          ['.' log|a| - log|b| = log|a/b|] 3) dx/(x2 + a2) Let x = a tan Î¸ On differentiating both sides, we get dx = a sec2 Î¸ dÎ¸ âˆ´ âˆ« dx/(x2+a2) = âˆ« a sec2 Î¸ / (a2 tan2Î¸ + a2) = âˆ« a sec2Î¸ dÎ¸ / a2 (tan2Î¸ + 1) = âˆ« a sec2Î¸ dÎ¸ / a2 sec2Î¸            ['.' sec2Î¸ - tan2Î¸ = 1] = 1/a âˆ« dÎ¸ = 1/a . Î¸ + C = 1/a tan-1 x/a    [x = a tan Î¸ â‡’ Î¸ = tan-1 x/a] âˆ«dx/(x2 + a2) = 1/a . tan-1 x/a

#### Summary

Formulae of integrals: âˆ«dx/(x2 - a2) = 1/2a . log |(x - a)/(x + a)| + C âˆ«dx/(a2 - x2) = 1/2a . log |(a - x)/(a + x)| + C âˆ«dx/(x2 + a2) = 1/a . tan-1 x/a + C 1) âˆ«1/(x2 - a2) dx = âˆ«1/((x + a)(x - a)) dx âˆ«1/((x + a)(x - a)) dx = 1/2a . âˆ« [1/(x-a) - 1/(x+a)] dx = 1/2a . âˆ« [1/(x-a)  dx- 1/(x+a) dx] = 1/2a . [log |x-a| - log |x+a|] + C = 1/2a . log|(x-a)/(x+a)| + C  ['.' log|a| - log|b| = log|a/b|] 2) âˆ« 1/(a2 - x2) dx âˆ« 1/((a+x)(a-x)) dx = 1/2a . âˆ«[1/(a+x) + 1/(a-x)] dx 1/2a [âˆ« 1/a+x dx + âˆ« 1/a-x dx] = 1/2a [log|a+x| - log|a-x|] + C = 1/2a log|(a+x)/(a-x)| + C          ['.' log|a| - log|b| = log|a/b|] 3) dx/(x2 + a2) Let x = a tan Î¸ On differentiating both sides, we get dx = a sec2 Î¸ dÎ¸ âˆ´ âˆ« dx/(x2+a2) = âˆ« a sec2 Î¸ / (a2 tan2Î¸ + a2) = âˆ« a sec2Î¸ dÎ¸ / a2 (tan2Î¸ + 1) = âˆ« a sec2Î¸ dÎ¸ / a2 sec2Î¸            ['.' sec2Î¸ - tan2Î¸ = 1] = 1/a âˆ« dÎ¸ = 1/a . Î¸ + C = 1/a tan-1 x/a    [x = a tan Î¸ â‡’ Î¸ = tan-1 x/a] âˆ«dx/(x2 + a2) = 1/a . tan-1 x/a

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