Integrals

Methods of Integration


TIPS FOR SOLVING INTEGRATING FUNCTIONS

  • Integrals are in the form
    \int \frac{1}{a \cos x + b \sin x + c} dx
  • Process :
    Put \tan \frac{x}{2} = t
    dx = \frac{2dt}{1 + t^{2}} 
    \cos x = \frac{1-t^{2}}{1+t^{2}}, \ \ \sin x = \frac{2t}{1 + t^{2}}
  • Integrals are in the form
    \int \frac{1}{a \cos 2x + b \sin 2x + c} dx
  • Process : Put tan x = t
    dx = \frac{dx}{1+t^{2}}
    \cos x = \frac{1-t^{2}}{1+t^{2}}, \ \ \sin x \frac{2t}{1 + t^{2}}
  • Integrals are in the form
    \int \frac{1}{a \cos^{2} x + b \sin^{2} x + c \sin x \cos x + d} dx
  • Process : Divide the numerator and denominator with cos2x then put k tan x = t (where k is coefficient of tan x)
  • Integrals which are in the form
    \int \frac{a \cos x + b \sin x}{c \cos x + d \sin x} dx
  • Process : Numerator = A \frac{d}{dx}(denominator) + B (denominator) then find the values of A and B by compare coefficients of cos x and sin x
  • Integrals which are in the form
    \int \frac{a \cos x + b \sin x + c}{d \cos x + e \sin x + f} dx
  • Process : Numerator = A \frac{d}{dx}(denominator) + B (denominator) + c then find the values of A and B by compare coefficients of cos x and sin x and constant terms.
  • Integrals which are in the form
    \int \frac{1}{(ax + b)\sqrt{px + q}} dx (or) \int \frac{ax+ b}{\sqrt{px + q}} dx (or) \int (ax + b) \sqrt{px + q} dx
  • Process : Put px + q = t2
    convert total integrand into t terms
  • Integrals which are in the form
    \int \frac{1}{\left(ax^{2}+b\right) \sqrt{px+q}} dx
  • Process : Put px + q = t2
    convert the total integrand into t terms
  • Integrals which are in the form
    \int \frac{1}{(px + q) \sqrt{ax^{2} + bx + c}} dx
  • Process : Put px + q = \frac{1}{t}
  • Convert the total integrand into t terms
  • Integrals which are in the form
    \int \frac{1}{(px^{2} + q) \sqrt{ax^{2} + b}} dx
  • Process :
    Put x = \frac{1}{t}
    Convert the total integrand into t terms

  • Important Tips

    • \int f^{'} (ax + b) dx = \frac{f(ax + b)}{a} + c
    • \int \frac{f^{'} (x)}{f(x)} dx = \log \mid f(x) \mid + c
    • \int \frac{f^{'} (x)}{\sqrt{f(x)}} dx = 2 \sqrt{f(x)} + c
    • \frac{d}{dx} \left(\int f(x) dx\right) = f(x)
    • \int \left(\frac{d}{dx} \left(f(x)\right)\right) dx = f(x) + c

Part1: View the Topic in this video From 45:12 To 52:19

Part2: View the Topic in this video From 00:40 To 55:14

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  1. ∫ tan x dx = log |sec x| + C
  2. ∫ cot x dx = log |sin x| + C
  3. ∫ sec x dx = log |sec x + tan x| + C
  4. ∫ cosec x dx = log |cosec x − cot x| + C
  5. \int \sin x dx = - \cos x + c
    \int \cos x dx = \sin x + c
    \int \tan x dx = \log \mid \sec x \mid + c \ \ (or)  - \log \mid \cos x \mid + c
    \int \cot x dx = \log \mid \sin x \mid + c \ \ (or)  - \log \mid cosex \ x \mid + c
    \int \sec x dx = \log \mid \sec x + \tan x \mid + c (or) \log \mid \tan \left(\frac{\pi}{4} + \frac{x}{2} \right) \mid + c
    \int cosec x \ dx = \log \mid cosec x - \cot x \mid + c (or) \log \mid \tan \frac{x}{2} \mid + c
  6. \int \sec^{2} x dx = \tan x + c
    \int cosec^{2} x dx = - \cot x + c
    \int \sec x \tan x dx = \sec x + c
    \int cosec x \cot x dx = - cosec x + c
  7. \int \sin^{-1}x dx = x \sin^{-1} x + \sqrt{1-x^{2}} + c
  8. \int \cos^{-1}x dx = x \cos^{-1} x - \sqrt{1-x^{2}} + c
  9. \int \tan^{-1}x dx = x \tan^{-1} x - \frac{1}{2} \log (1+x^{2})+ c
  10. \int \cot^{-1}x dx = x \cot^{-1} x + \frac{1}{2} \log (1+x^{2})+ c
  11. \int \sec^{-1}x dx = x \sec^{-1} x - \cos h^{-1} x + c
  12. \int cosec^{-1}x dx = x cosec^{-1} x + \cos h^{-1} x + c