# 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

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

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

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