Differentiation Rules Calculus I Integration Formulas Calculus I

Integration Formulas
Calculus I
Differentiation Rules
Calculus I
General Formulas
R
1. k dx = kx + C.
General Formulas
1.
d
dx k
2.
d
dx (u
+ v) =
du
dx
+
dv
dx .
2.
R
(u + v) dx =
R
u dx +
R
v dx.
3.
d
dx (u
− v) =
du
dx
−
dv
dx v.
3.
R
(u − v) dx =
R
u dx −
R
v dx.
4.
d
dx (uv)
4.
R
u dv = uv −
R
v du. (Integration by parts)
5.
d u
dx ( v )
= 0.
dv
= u dx
+ v du
dx .
=
dv
v du
dx −u dx
v2
.
6. y = f (u), u = u(x),
5. No “Quotient Rule”.
dy
dx
=
dy du
du dx .
Basic Functions
1.
d n
dx u
2.
d
dx
3.
d u
dx e
4.
d
dx
5.
=
nun−1 du
dx .
ln |u| =
1 du
u dx .
6.
R
f (u) du =
R
f (u(x))u0 (x) dx. (Substitution)
Basic Formulas
R
1
un+1 + C; n 6= −1.
1. un du = n+1
2.
R
1
u
3.
R
eu du = eu + C.
sin u = cos u du
dx .
4.
R
sin u du = − cos u + C.
d
dx
cos u = − sin u du
dx .
5.
R
cos u du = sin u + C.
6.
d
dx
tan u = sec2 u du
dx .
6.
R
sec2 u du = tan u + C.
7.
d
dx
sec u = sec u tan u du
dx .
7.
R
sec u tan u du = sec u + C.
8.
d
dx
cot u = − csc2 u du
dx .
8.
R
csc2 u du = − cot u + C.
9.
d
dx
csc u = − csc u cot u du
dx .
9.
R
csc u cot u du = − csc u + C.
10.
d
dx
ln | sec u| = tan u du
dx .
10.
R
tan u du = ln | sec u| + C.
11.
d
dx
ln | sec u + tan u| = sec u du
dx .
11.
R
sec u du = ln | sec u + tan u| + C.
12.
d
dx
sinh u = cosh u du
dx .
12.
R
sinh u du = cosh u + C.
d
dx
cosh u = sinh u du
dx .
13.
R
13.
cosh u du = sinh u + C.
14.
R
sech2 u du = tanh u + C.
15.
R
√ 1
a2 −u2
16.
R
1
a2 +u2
17.
R
√ 1
u u2 −a2
18.
R
√ 1
a2 +u2
19.
R
1
a2 −u2
14.
d
dx
15.
d
dx
16.
d
dx
17.
d
dx
18.
d
dx
19.
d
dx
= eu du
dx .
tanh u = sech2 u du
dx .
−1
sin
−1
tan
sec
−1
u=
u=
1 du
1+u2 dx .
u=
du
√1
.
u u2 −1 dx
−1
sinh
−1
tanh
du
√ 1
.
1−u2 dx
u=
u=
du
√ 1
.
1+u2 dx
1 du
1−u2 dx .
du = ln |u| + C.
du = sin−1
du =
1
a
u
a
tan−1
du =
1
a
1
a
u
a
+ C.
sec−1
du = sinh−1
du =
+ C.
u
a
tanh−1
u
a
+ C.
+ C.
u
a
+ C. if u2 < a2 .