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 .