icon-komplexe-Zahl Math Tutorial: Integral Calculations

Basic Integrals

General

fxdx = fx dx = x

Power

xndx = xn+1n+1 ax+bndx = ax+bn+1an+1 dxx = ln|x|

Exponential Functions

exdx = ex axdx = axlna

Trigonometric Functions

sinxdx = -cosx cosxdx = sinx tanxdx = -ln|cosx| dxcos2x = tanx dxsin2x = -cotx

Logarithmic Functions

lnxdx = xlnx-x lnx2dx = xlnx2 -2xlnx+2x dxxlnx = lnlnx

Irrational Functions

ax+bdx = 23aax+b3 dxax+b = 2ax+ba

Calculation Rules for undefined Integrals

a) Additivity A sum under the integral is integrated by the summands are integrated separately and then summed.

fx+gxdx = fxdx+gxdx

b) Factor Rule A constant factor a can be pulled out of the integral.

afxdx = a fxdx

c) If F(u) is the antiderivative of f(u), then the following relationship holds for arbitrary constants a and b with a not equal to 0.

fax+bdx = 1a Fax+b+C

d) If f(x) is differentiable and f(x) not equal to 0, then the following relationship applies.

f xfxdx = lnfx+C

e) Integration by parts Are the functions u(x) and v(x) are differentiable, then the following relationship holds.

uxvxdx = uxvx-uxvxdx

f) Substitution If the function f(z) is continuous and z = g(x) is differentiable, then the following relationship holds.

fgxgxdx = fzdz

Examples

Example: Linearity of the integral (Rule a and b)

5x+2sinxdx

= 5xdx+2sinxdx

Ⅰ:

Due to the additivity of the summands can be integrated individually.

= 5xdx+2sinxdx

Ⅱ:

The constant factors are taken outside the integral.

= 52x2-2cosx+C

Ⅲ:

With the basic integrals, the solution follows.

Example: A function of a linear function (Rule c)

12x+3dx

fax+b = 12x+3

Ⅰ:

The function under the integral is a function of a linear function.

Fu=1udu=ln|u|+C

Ⅱ:

Formation of the primitive of f with the substitution u = ax + b.

= 12ln|2x+3|+C

Ⅲ:

Install in accordance with rule c yields the solution of the integral.

Example: f(x) in the denominator and the derivative of f(x) in the nominator (Rule d)

cosxsinxdx

ddxsinx = cosx

Ⅰ:

That means in the numerator is the derivative of the denominator.

= ln|sinx|+C

Ⅱ:

Install in accordance with rule d yields the solution of the integral.

Example: Integration by parts (Rule e)

x3lnxdx

x3dx = x44

Ⅰ:

Is the first factor v', then the integral of the first factor is v.

ddxlnx = 1x

Ⅱ:

The derivative of the second factor is u'.

= x44lnx-14x3dx

Ⅲ:

Put in following rule e.

= x44lnx-14x44+C

Ⅳ:

With the integral of I the solution of the integral follows.

= x44lnx-14+C

Ⅴ:

This can be written also shorter by factoring out.

Example: Integration by substitution (Rule f)

xdxa2+x2

gx = a2+x2

Ⅰ:

Substitution of the denominator as a function g(x).

dgdx = xa2+x2 = xg

Ⅱ:

The derivative of g(x) by dx is the relationship according to the differentials.

xdx = gdg

Ⅲ:

Dissolved, the term can be used in the integral.

= gdgg = dg = g+C

Ⅳ:

Put in and integrated.

= g+C = a2+x2+C

Ⅴ:

Repatriation of substitution yields the solution.

Rules of calculation of definite integrals

The existence of the integrals in each case provided. There are a, b, c ∈ ℜ constants.

a) Interchange of the limits of integration.

abfxdx = -bafxdx

a1) Definition

aafxdx = 0

b) Factor Rule A constant factor a can be pulled outside the integral.

abafxdx = a abfxdx

c) Sum Rule A sum under the definite integral can be integrated by means of the summands.

abfx+gxdx = abfxdx+abgxdx

d) Decomposition of the definite integral in part integrals.

abfxdx = acfxdx+cbfxdx

e) Mean Value Theorem If f is integrable and we have m ≤ f ≤ M, then there exists at least one number μ with m ≤ μ ≤ M and we have the following relationship.

abfxdx = μb-a

f) General Mean Value Theorem If f and g are integrable and we have m ≤ f ≤ M and g ≥ 0 either always or g ≤ 0, then there exists at least one number μ with m ≤ μ ≤ M and it works as follows.

abfxgxdx = μabgxdx

g) Second Mean Value Theorem If f is monotonic and bounded and g integrable, then there exists at least one number μ for which the following relationship is valid.

abfxgxdx = faaμgxdx+fbμbgxdx

h) If a function f is continuous and differentiable, then applies

Fx = axftdt

with Fx = fx

i) Main Theorem of the Differential and Integral Calculations. If the antiderivative F of f is known, then the definite integral of f is calculated as follows

abfxdx = Fb-Fa

j) Integration by parts Are the functions u (x) and v (x) is differentiable, then the following relationship holds.

abuxvxdx = uxvxab-abuxvxdx

k) Substitution for definite integrals.

abfgxgxdx = gagbfzdz