Integration Strategy, Exercises of Mathematics

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Integration strategy

Phan Phuong Dung

August 21, 2023

(^1) 7.1-7.5. Integration strategy Substitution Integration by parts Table of integrals

(^2) 7.6. First-Order Linear differential equations Definition and general solution Applications of First-Order Equations Modelling Dilution

(^3) 7.7. Improper integrals

7.1-7.5. Integration strategy Substitution

Remark When performing substitution to find definite integrals, we must change the integration limits.

Example

Find

Z 1

0

ex e^2 x^ + 4dx.

7.1-7.5. Integration strategy Integration by parts

Integration by parts

Formula for integration by parts Z udv = uv −

Z

vdu

Z (^) b

a

udv = uv |ba −

Z (^) b

a

vdu.

Example

Find

Z

xe^2 x+1dx.

7.1-7.5. Integration strategy Integration by parts

Integration by parts

When do we use Integration by parts? Some cases a. Forms

R

xneax^ dx,

R

xnsinaxdx,

R

xncosaxdx; let u = xn. b. Forms

R

xnlnxdx,

R

xnsin−^1 axdx,

R

xntan−^1 axdx; let dv = xndx. c. Forms

R

eax^ sinbxdx,

R

eax^ cosbxdx; let dv = eax^ dx.

7.1-7.5. Integration strategy Integration by parts

Integration of trigonometric functions

R

sinmxcosnxdx m odd: let u = cos x. n odd: let u = sin x. m and n both even: use half-angle identities cos^2 x = 12 (1 + cos(2x) and sin^2 x = 12 (1 − cos 2x). R tanmxsecnxdx n even: let u = tanx. m odd: let u = secx.

7.1-7.5. Integration strategy Integration by parts

Solution.

7.1-7.5. Integration strategy Integration by parts

Integration of radical functions

Radical integral a. Form

a^2 − u^2 : let u = a sin θ. b. Form

a^2 + u^2 : let u = a tan θ. c. Form

u^2 − a^2 : let u = a sec θ.

Example Find

Z √

9 − x^2 dx.

7.1-7.5. Integration strategy Integration by parts

Solution.

7.1-7.5. Integration strategy Table of integrals

Table of integrals

Appendix D Elementary forms (Formulas 1-29) Linear and quadratic forms (Formulas 30-76): au + b, u^2 + a^2 , u^2 − a^2 , au^2 + bu + c. Radical forms (Formulas 77-121):

au + b,

√ u^2 +^ a^2 , u^2 − a^2 ,

a^2 − u^2. Trigonometric forms (Formulas 112 - 167): cos au, sin au, both sin au and cos au, tan au, cot au, sec au, csc au. Inverse trigonometric forms (Formula 168-182) Exponential and logarithmic forms (Formulas 183- 200)

7.1-7.5. Integration strategy Table of integrals

Solution.

7.1-7.5. Integration strategy Table of integrals

General strategy for integration

Try the table of integrals When working with rational integral, try decomposition method Try integration by parts Try institution Try rewriting the integral, start the steps above again

7.6. First-Order Linear differential equations Definition and general solution

Example Is the differential equation

x^2 dydx − (x^2 + 2)y = x^5

first-order linear?

7.6. First-Order Linear differential equations Definition and general solution

Theorem 7.1 General solution of a first-order LDE The general solution of the first-order linear differential equation

dy dx +^ P(x)y^ =^ Q(x)

is given by

y = (^) I (^1 x)

Z

I (x)Q(x)dx + c

where I (x) = e

R (^) P(x)dx , the exponent is any antiderivative of P(x), and C is arbitrary constant.

I (x) is called an integrating factor of the DE.