Chp 9: Integration 2

Question 1

Integral of sin x ^n

Answer 1

-( 1/n ) sin x ^ (n-1) cos x + (n-1)/n integral of sin x^ (n-2) dx

Question 2

Integral of cos x^ n

Answer 2

( 1/n ) cos x ^ (n-1) sin x + (n-1)/n integral of cos x^ (n-2) dx

Question 3

Integral of (tan x)^n

Answer 3

( (Tan x )^n-1 / n - 1 ) - integral of (tan x) ^(n-2)

Question 4

Integral of ( sec x )^n

Answer 4

( Sec x ^ (n-2) tan x / n-1 ) + (n-2)/(n-1) integral of sec x^n-2

Question 5

Cos ^n (odd)

Answer 5

1. Split into factors of cos x
2. Apply cos^2 x = 1- sin ^2 x
3. Sub u = sin x

Question 6

Sin ^m x (odd)

Answer 6

1. Split into factors of sin x
2. Apply sin^2 x = 1- cos ^2 x
3. Sub u = cos x

Question 7

Sin ^ m x cos ^ n x (even)

Answer 7

Use sin^2 x = 1/2(1-cos 2x) and cos^2 x = 1/2(1+ cos 2x)

Question 8

Integral of 1/(x^2 - a^2)

Answer 8

1/2a (ln (x-a/x+a) ) + C

Question 9

Integral of 1/(x^2 + a^2)

Answer 9

1/a ( tan ^-1 (x/a) ) + c

Question 10

Integral of 1/(a^2 - x^2 )

Answer 10

1/2a ( ln (a + x/ a-x ) ) + c

Question 11

Integral of 1/ [(a^2 - x^2 )]^1/2

Answer 11

Inverse sin (x/a) + C

Question 12

Integral of 1/(x^2 + a^2)^1/2

Answer 12

Inverse sinh (x/a) + C

Question 13

Integral of 1/(x^2 - a^2)^1/2

Answer 13

Inverse cosh (x/a) + C

Question 14

Integral of u dv

Answer 14

uv - integral of v du

Question 15

Sec ^n x (even)

Answer 15

1. Split into factors of sec^2 x
2. Apply sec^2 x = tan ^2 x + 1
3. Sub u = tan x

Question 16

Tan ^m x ( odd)

Answer 16

1. Split into factors of sec x tan x
2. Apply tan^2 x = sec^ 2 x - 1
3. Sub u = sec x

Question 17

Tan^m x (even) and sec ^n x (odd)

Answer 17

1. Use tan^2 x = sec^ 2 x - 1 to make into sec x

2. Use reduction formula for sec x