Trigonometric Integrals

Question 1

∫sin(ax)sin(bx) = ?

Answer 1

∫1/2cos((a − b)x) − 1/2cos((a + b)x)

Question 2

∫sin(ax)cos(bx) = ?

Answer 2

∫ 1/2sin((a-b)x) + 1/2sin((a+b)x)

Question 3

∫cos(ax)cos(bx) = ?

Answer 3

∫ 1/2cos((a-b)x) + 1/2cos((a+b)x)

Question 4

∫ sec²x dx = ?

Answer 4

tanx + C

Question 5

∫ sec x tanx dx = ?

Answer 5

sec x + C

Question 6

∫ tanx dx = ?

Answer 6

ln|sec x| + C

Question 7

∫ sec x dx = ?

Answer 7

ln|sec x + tanx| + C.

Question 8

Reduction*
∫ sec^nx dx = ?

Answer 8

(1/n-1)sec^(n-2)xtanx + (n-2/n-1) ∫ sec^(n-2)dx

Question 9

Reduction*
∫ tan^nx dx = ?

Answer 9

(1/n-1)tan^(n-1)x - ∫ tan^(n-2)dx

Question 10

given ∫cos^j x * sin^kx

Answer 10

1. if k = odd, write sin^kx= sin^(k-1)sinx
2. if j = odd, write cos^jx = cos^(j-1)cosx
3. if both k j are even use identity

Question 11

given ∫tan^k x * sec^j x

Answer 11

1. if j = even, write sec^jx= sec^(j-2)sec^2x
2. if k = odd, write tan^(k-1)x*sec^(j-1)xsecxtanx