Calculus 2
2nd Semester
Derivatives of Trigonometric Functions un+1
• ∫ un du = + c ;n = -1
n+1
• sin u = cos u du 1
• ∫ u du = ln|u| + c
• cos u = - sin u du
• tan u = sec2u du au
• ∫ au du= +c
• sec u = sec u tan u du ln a
• csc u = - csc u cot u du
• ∫ eu du = eu + c
• cot u = - csc2u du
Pythagorean Relations • ∫ sin u du = - cos u + c
• sin2x + cos2x = 1 • ∫ cos u du = sin u + c
✷ sin2x = 1 - cos2x
• ∫ tan u du = ln | sec u | + c
✷ cos2x = 1 - sin2x
• tan2x + 1 = sec2x
• ∫ sec u du = ln | sec u + tan u | + c
✷ tan2x = sec2x – 1
• cot2x + 1 = csc2x
• ∫ csc u du = - ln | csc u + cot u | +c
✷ cot2x = csc2x – 1
Quotient Relations • ∫ cot u du = ln | sin u | + c
sin x cos x • ∫ sec2 u du = tan u + c
• tan x = • cot x =
cos x sin x
• ∫ csc2 u du = - cot u + c
Reciprocal Relations
• ∫ sec u tan u du = sec u + c
1 1
• sin x= • csc x =
csc x sin x • ∫ csc u cot u du = - csc u + c
1 1
• cos x= • sec x = du u u
sec x cos x • ∫√ = arcsin a + c or -arcos +c
a2 - u2 a
1 1
• tan x= • cot x = du 1 u 1 u
cot x tan x • ∫ 2 = a arctan a + c or - a arcot +c
a2+u a
Integration Rules • ∫
du 1 u 1
= a arcsec a + c or - a arcsc
u
+c
u√a2 - u2 a
• ∫ dx = x + c
Cases
• ∫ kdx = k ∫ dx
• Case 1: ∫ sinm u cosn u du
xn+1
• ∫ x2 dx = +c ✷ where m and/or n are odd
n+1
✷ ∫ sinm-1 cosn sin u du
• ∫ f(x1 ) + f(x2 ) dx = ∫ fx1 dx+ ∫ fx2 dx
Calculus 2
2nd Semester
• Case 2: ∫ sinm u cosn u du
✷ where m and n are even
1 + cos 2x
✷ if: cos2 x =
2
1 - cos 2x
✷ if: sin2 x = 2
• Case 3: ∫ sin mx cos nx dx
✷ m > n: sin mx cos nx
1
= 2 [sin (m+n)x + sin (m–n)x]
✷ n > m: sin mx cos nx
1
= 2 [sin (m+n)x - sin (n–m)x]
✷ cos mx cos nx
1
= 2 [cos (m+n)x + sin (m–n)x]
✷ sin mx cos nx
1
= 2 [cos (m+n)x - cos (m–n)x]
• Case 4: ∫ secm udu or ∫ cscn udu
✷ where m and n are even
✷ ∫ secm-2 u sec2 u du
✽ substitute: tan2x + 1 = sec2x
✷ ∫ cscn-2 u csc2 u du
✽ substitute: cot2x + 1 = csc2x
• Case 5: ∫ tanm udu or ∫ cotn udu
✷ ∫ tanm-2 u tan2 u du
✽ substitute: tan2x = sec2x – 1
✷ ∫ cotm-2 u cot2 u du
✽ substitute: cot2x = csc2x – 1
• Case 6: ∫ tanm u secn u du or
or ∫ cotm u cscn u du or
