Calculus (I)

Calculus (I)

W^EN-C^HING L^IEN

Department of Mathematics National Cheng Kung University

2008

8.3 Tables of Integrals

(1) Basic Functions (2) Rational Functions (3) Square Roots

(4) Trigonometric Functions (5) Exponential Functions (6) Logarithmic Functions

8.3 Tables of Integrals

(1) Basic Functions (2) Rational Functions (3) Square Roots

(4) Trigonometric Functions (5) Exponential Functions (6) Logarithmic Functions

8.3 Tables of Integrals

(1) Basic Functions (2) Rational Functions (3) Square Roots

(4) Trigonometric Functions (5) Exponential Functions (6) Logarithmic Functions

8.3 Tables of Integrals

(1) Basic Functions (2) Rational Functions (3) Square Roots

(4) Trigonometric Functions (5) Exponential Functions (6) Logarithmic Functions

8.3 Tables of Integrals

(1) Basic Functions (2) Rational Functions (3) Square Roots

(4) Trigonometric Functions (5) Exponential Functions (6) Logarithmic Functions

8.3 Tables of Integrals

(1) Basic Functions (2) Rational Functions (3) Square Roots

(4) Trigonometric Functions (5) Exponential Functions (6) Logarithmic Functions

Examples:

1

Z p

9+4x²dx

2

Z

(x +1)²e^−2xdx

3

Z

cos²(5x−3)dx

4

Z

e^2x+1sin(πx 2 )dx

Examples:

1

Z p

9+4x²dx

2

Z

(x +1)²e^−2xdx

3

Z

cos²(5x−3)dx

4

Z

e^2x+1sin(πx 2 )dx

Examples:

1

Z p

9+4x²dx

2

Z

(x +1)²e^−2xdx

3

Z

cos²(5x−3)dx

4

Z

e^2x+1sin(πx 2 )dx

Examples:

1

Z p

9+4x²dx

2

Z

(x +1)²e^−2xdx

3

Z

cos²(5x−3)dx

4

Z

e^2x+1sin(πx 2 )dx

Examples:

1

Z p

9+4x²dx

2

Z

(x +1)²e^−2xdx

3

Z

cos²(5x−3)dx

4

Z

e^2x+1sin(πx 2 )dx

Example:

Integration of R(x,√

1−x²)

By using the substitution x =cos u ⇒

√

1−x²=sin u dx =−sin udu (or x =sin u)

Example:

Integration of R(x,√

1−x²)

By using the substitution x =cos u ⇒

√

1−x²=sin u dx =−sin udu (or x =sin u)

Example:

Integration of R(x,√

1−x²)

By using the substitution x =cos u ⇒

√

1−x²=sin u dx =−sin udu (or x =sin u)

Integral Tables

Z 1

√a²−x²dx =sin⁻¹ x a +c Z 1

a²+x²dx = 1

atan⁻¹x a +c Z √

u²−a²

u du=p

u²−a²−a sec⁻¹u a +c

Z du

u²(a+bu) =− 1 au + b

a² ln

a+bu u

+c Z du −√

a²+u²

Integral Tables

Z 1

√a²−x²dx =sin⁻¹ x a +c Z 1

a²+x²dx = 1

atan⁻¹x a +c Z √

u²−a²

u du=p

u²−a²−a sec⁻¹u a +c

Z du

u²(a+bu) =− 1 au + b

a² ln

a+bu u

+c Z du −√

a²+u²

Integral Tables

Z 1

√a²−x²dx =sin⁻¹ x a +c Z 1

a²+x²dx = 1

atan⁻¹x a +c Z √

u²−a²

u du=p

u²−a²−a sec⁻¹u a +c

Z du

u²(a+bu) =− 1 au + b

a² ln

a+bu u

+c Z du −√

a²+u²

Integral Tables

Z 1

√a²−x²dx =sin⁻¹ x a +c Z 1

a²+x²dx = 1

atan⁻¹x a +c Z √

u²−a²

u du=p

u²−a²−a sec⁻¹u a +c

Z du

u²(a+bu) =− 1 au + b

a² ln

a+bu u

+c Z du −√

a²+u²

Integral Tables

Z 1

√a²−x²dx =sin⁻¹ x a +c Z 1

a²+x²dx = 1

atan⁻¹x a +c Z √

u²−a²

u du=p

u²−a²−a sec⁻¹u a +c

Z du

u²(a+bu) =− 1 au + b

a² ln

a+bu u

+c Z du −√

a²+u²

Integral Tables

Z 1

√a²−x²dx =sin⁻¹ x a +c Z 1

a²+x²dx = 1

atan⁻¹x a +c Z √

u²−a²

u du=p

u²−a²−a sec⁻¹u a +c

Z du

u²(a+bu) =− 1 au + b

a² ln

a+bu u

+c Z du −√

a²+u²

Examples:

1

Z 1

4+x²+2xdx

2

Z du 2u²√

1+4u²

3

Z √

3x²−1

2x dx

Examples:

1

Z 1

4+x²+2xdx

2

Z du 2u²√

1+4u²

3

Z √

3x²−1

2x dx

Examples:

1

Z 1

4+x²+2xdx

2

Z du 2u²√

1+4u²

3

Z √

3x²−1

2x dx

Examples:

1

Z 1

4+x²+2xdx

2

Z du 2u²√

1+4u²

3

Z √

3x²−1

2x dx

Thank you.