Miscellaneous

Chapter 7 Class 12 Integrals
Serial order wise

Check the formula sheet of integration.

Topics include

1. Basic Integration Formulas
2. Integral of special functions
3. Integral by Partial Fractions
4. Integration by Parts
5. Other Special Integrals
6. Area as a sum
7. Properties of definite integration

Integration of Trigonometric Functions, Properties of Definite Integration are all mentioned here.

## Basic Formula

1. ∫x  = x n+1 /n+1  + C
2. ∫cos x    = sin x  + C
3. ∫sin x    = -cos x  + C
4. ∫sec 2 x    = tan x  + C
5. ∫cosec 2 x    = -cot x  + C
6. ∫sec x tan x    = sec x  + C
7. ∫cosec  x cot x    = -cosec x  + C
8. ∫dx/√ 1- x 2  = sin -1  x  + C
9. ∫dx/√ 1- x 2  = -cos -1  x  + C
10. ∫dx/√ 1+ x 2  = tan -1  x  + C
11. ∫dx/√ 1+ x 2  = -cot -1  x  + C
12. ∫e  = e + C
13. ∫a  = a x / log a + C
14. ∫dx/x √ x 2   - 1= sec -1  x  + C
15. ∫dx/x √ x 2   - 1= cosec -1  x  + C
16. ∫1/x    = log |x| + c
17. ∫tan x    = log |sec x| + c
18. ∫cot x    = log |sin x| + c
19. ∫sec x    = log |sec x + tan x| + c
20. ∫cosec x    = log |cosec x - cot x| + c

## Integrals of some special function s

1.  ∫dx/(x 2   - a 2 ) = 1/2a  log⁡ |(x - a) / (x + a)| + c
2.  ∫dx/(a 2   - x 2 ) = 1/2a  log⁡ |(a + x) / (a - x)| + c
3. ∫dx / (x 2   + a 2 ) = 1/a  tan (-1) ⁡ x / a + c
4. ∫dx / √(x 2   - a 2 ) = log |"x" + √(x 2 -a 2 )| + C

5. 1.∫dx / √(a 2   - x 2 ) = sin-1 x / a + c

6. ∫dx / √(x 2 + a 2 ) = log |"x" + √(x 2 + a 2 )| + C

Check Practice Questions

## Integrals by partial fractions

1. (px + q) / ((x - a) (x - b)) = A/(x - a) + B / (x - b)

2. (px + q) / (x - a) 2  = A/(x - a) + B / (x - a) 2

3. (px 2   + qx + r) / (x - a) (x - b) (x - c)  = A / (x - a) + B / (x - b) + C / (x - c)
4. (px 2 + qx + r) / ((x - a) 2 (x - b) ) = A / (x - a) + B / (x - a) 2 + C / (x - b)
5. (px 2 + qx + r) / (x - a) (x 2 + bx + c)  = A / (x - a) + (Bx + C) / (x 2 + bx + c)

Where x 2 + bx + c can not be factorised further.

Check Practice Questions

## Integration by parts

1. ∫𝒇(𝒙) 𝒈⁡(𝒙)  𝒅𝒙 = 𝒇(𝒙) ∫𝒈 (𝒙) 𝒅𝒙− ∫(𝒇 ' (𝒙) ∫𝒈(𝒙) 𝒅𝒙) 𝒅𝒙

To decide first function. We use

I → Inverse (Example sin (-1)  ⁡x)

L → Log (Example log ⁡x)

A → Algebra (Example x 2 , x 3 )

T → Trigonometry (Example sin 2 x)

E → Exponential (Example e x )

2. ∫ex [f (x) + f ′(x)] dx = ∫ex f(x) dx + C

Check Practice Questions

## Other Special Integrals

1.  ∫√ (x - a 2 ) dx = x / 2 √(x - a 2 ) − a / 2 log |x + √(x - a 2 )| + C
2. √( x + a 2 ) dx = x / 2 √(x + a 2 ) + a / 2 log |x +√(x + a 2 )| + C

3. √( a - x 2 ) dx = x / 2 √(a 2   - x 2 ) + a / 2 sin 1 x / a + C

Check Practice Questions

## Integral of the form  ∫ (px+q) √( ax 2  + bx + c )  dx

We solve this using a specific method.

1. First we write
px + q = A (d(√(ax + bx + c))/dx) + B
2. Then we find A and B
3. Our equation becomes two seperate identities and then we solve.

Some examples are

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## Area as a sum

∫a→b f (x)  dx = (b - a)  (lim) (n→∞)  1 / n (f (a) + f (a + h) + f (a + 2h)…+ f (a + (n - 1) h))

Check Practice Questions

## Properties of definite integration

1. P 0 : ∫a→b   f(x) dx = ∫a→b   f(t) dt
2. P 1 : ∫a→b   f(x) dx = -∫b→a   f(x) dx .In particular, ∫a→a   f(x) dx = 0
3. P 2 : ∫a→b   f(x) dx = ∫a→c f(x) dx + ∫c→b f(x) dx
4. P 3 : ∫a→b f(x) dx= ∫a→b   f(a + b - x) dx.
5. P 4 : ∫0→a f(x)dx = ∫0→a   f(a - x) dx
6. P 5 : ∫0→2a   f(x) dx = ∫0→a   f(x) dx + ∫0→a f(2a - x) dx
7. P 6 :  ∫0→2a f(x) = {(2∫0→a   f(x) dx,  if f (2a - x) = f (x) , if f (2a - x) = -f(x))
8. P 7 :  ∫(-a)→a f(x) = {(2∫0→a f(x) dx,  if f(-x) = f(x), if f ( -x) = -f(x)

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