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Integral long logarithm proof. Solution of logarithms. Examples. Properties of logarithms. What is a logarithm

What is a logarithm?

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very ..."
And for those who are "very even ...")

What is a logarithm? How do you solve logarithms? These questions confuse many graduates. Traditionally, the topic of logarithms is considered difficult, incomprehensible and scary. Especially - equations with logarithms.

This is absolutely not the case. Absolutely! Don't believe me? Good. Now, in some 10 - 20 minutes, you:

1. Understand what is logarithm.

2. Learn to solve a whole class of exponential equations. Even if you haven't heard of them.

3. Learn to calculate simple logarithms.

And for this you will only need to know the multiplication table, but how a number is raised to a power ...

I feel you are in doubt ... Well, watch the time! Go!

Start by solving the following equation in your head:

If you like this site ...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Instant validation testing. Learning - with interest!)

you can get acquainted with functions and derivatives.

Table of antiderivatives ("integrals"). Integral table. Tabular indefinite integrals. (The simplest integrals and integrals with a parameter). Integration formulas by parts. Newton-Leibniz formula.

Table of antiderivatives ("integrals"). Tabular indefinite integrals. (The simplest integrals and integrals with a parameter).
Integral of a power function.	Integral of a power function.
An integral that reduces to an integral of a power function if x is driven under the sign of the differential.
	The integral of the exponent, where a is a constant number.
Integral of a complex exponential function.	Integral of an exponential function.
Integral equal to the natural logo.	Integral: "Long logarithm".
	Integral: "Long logarithm".
Integral: "High logarithm".	The integral, where x in the numerator is entered under the sign of the differential (the constant under the sign can be either added or subtracted), in the end is similar to the integral equal to the natural logo.
Integral: "High logarithm".
Integral of the cosine.	Sine integral.
Integral equal to the tangent.	Integral equal to the cotangent.
Integral equal to both arcsine and arccosine
Integral equal to both the inverse sine and the inverse cosine.	Integral equal to both arc tangent and arc cotangent.
Integral equal to cosecant.	Integral equal to secant.
Integral equal to the arcsecant.	Integral equal to the arcsecant.
Integral equal to the arcsecant.	Integral equal to the arcsecant.
Integral equal to the hyperbolic sine.	Integral equal to the hyperbolic cosine.
Integral equal to the hyperbolic sine, where sinhx is the hyperbolic sine in the English version.	Integral equal to the hyperbolic cosine, where sinhx is the hyperbolic sine in the English version.
Integral equal to the hyperbolic tangent.	Integral equal to the hyperbolic cotangent.
Integral equal to the hyperbolic secant.	Integral equal to the hyperbolic cosecant.

Integration formulas by parts. Integration rules.

Integration formulas by parts. Newton-Leibniz formula. Integration rules.
Integration of the product (function) by a constant:
Integration of the sum of functions:
indefinite integrals:
Integration by parts formula definite integrals:
Newton-Leibniz formula definite integrals:	Where F (a), F (b) are the values ​​of antiderivatives at points b and a, respectively.

Derivatives table. Tabular derivatives. Derivative of the work. Derivative of the quotient. Derivative of a complex function.

If x is an independent variable, then:

Derivatives table. Table derivatives. "Table derived" - yes, unfortunately, this is how they are searched for on the Internet
	Derivative of a power function
	Exponent derivative
Derivative of a complex exponential function	Derivative of an exponential function
Derivative of a logarithmic function	Derivative of the natural logarithm
	Derivative of the natural logarithm of the function
Sine derivative	Derivative of the cosine
The cosecant derivative	Secant derivative
Arcsine derivative	Derivative of the arccosine
Arcsine derivative	Derivative of the arccosine
Derivative of the tangent	Derivative of the cotangent
Derivative of the arctangent	Derivative of the arc cotangent
Derivative of the arctangent	Derivative of arc cotangent
Derivative of the arcsecant	Derivative of arcsecant
Derivative of the arcsecant	Derivative of arcsecant
Derivative of the hyperbolic sine Derivative of the hyperbolic sine in the English version	Derivative of the hyperbolic cosine Derivative of the hyperbolic cosine in the English version
Derivative of the hyperbolic tangent	Derivative of the hyperbolic cotangent
Derivative of hyperbolic secant	Derivative of the hyperbolic cosecant

Differentiation rules. Derivative of the work. Derivative of the quotient. Derivative of a complex function.
Derivative of the product (function) by a constant:
Derivative of the sum (functions):
Derivative of the product (functions):
Derivative of the quotient (functions):
Derivative of a complex function:

Properties of logarithms. Basic formulas for logarithms. Decimal (lg) and natural logarithms (ln).

Basic logarithmic identity
Let us show how it is possible to make any function of the form a b exponential. Since a function of the form ex is called exponential, then
Any function of the form a b can be represented as a power of ten

Natural logarithm ln (logarithm base e = 2.718281828459045 ...) ln (e) = 1; ln (1) = 0

Taylor series. Decomposition of a function in a Taylor series.

It turns out that most practically occurring mathematical functions can be represented with any accuracy in the vicinity of some point in the form of power series containing the degrees of the variable in ascending order. For example, in the vicinity of the point x = 1:

When using rows called by the ranks of Taylor, mixed functions containing, say, algebraic, trigonometric and exponential functions can be expressed as purely algebraic functions. Series can often be used to quickly differentiate and integrate.

The Taylor series in the vicinity of point a has the following forms:

1) , where f (x) is a function having derivatives of all orders for x = a. R n - the remainder of the Taylor series is determined by the expression

2)

The k-th coefficient (at x k) of the series is determined by the formula

3) A special case of the Taylor series is the Maclaurin (= McLaren) series (decomposition occurs around the point a = 0)

for a = 0

the members of the series are determined by the formula

Conditions for the application of the Taylor series.

1. For the function f (x) to be expanded into a Taylor series on the interval (-R; R), it is necessary and sufficient that the remainder in the Taylor (Maclaurin (= McLaren)) formula for this function tends to zero at k → ∞ on the indicated interval (-R; R).

2. It is necessary that there are derivatives for the given function at the point in the vicinity of which we are going to construct the Taylor series.

Properties of Taylor series.

If f is an analytic function, then its Taylor series at any point a of the domain of f converges to f in some neighborhood of a.

There are infinitely differentiable functions whose Taylor series converges but differs from a function in any neighborhood of a. For example:

Taylor series are used for approximation (approximation - scientific method, consisting in replacing some objects with others, in one sense or another close to the original, but simpler) functions by polynomials. In particular, linearization ((from linearis - linear), one of the methods of approximate representation of closed nonlinear systems, in which the study of a nonlinear system is replaced by an analysis of a linear system, in a sense equivalent to the original one.) Equations occurs by expanding into a Taylor series and cutting off all the terms above first order.

Thus, almost any function can be represented as a polynomial with a given accuracy.

Examples of some widespread expansions of power functions in the Maclaurin series (= McLaren, Taylor in the vicinity of point 0) and Taylor in the vicinity of point 1. The first terms of the expansions of the main functions in the Taylor and McLaren series.

Examples of some common expansions of power functions in Maclaurin series (= McLaren, Taylor in the vicinity of point 0)

Examples of some common Taylor series expansions in the vicinity of point 1

Antiderivatives table.

The properties of the indefinite integral make it possible to find its antiderivative from the known differential of a function. Thus, using equalities and it is possible to compose a table of antiderivatives from the table of derivatives of basic elementary functions.

Recall derivatives table, we also write it in the form of differentials.

For example, let us find the indefinite integral of the power function.

Using the table of differentials , therefore, by the properties of the indefinite integral, we have. therefore or in another post

Let us find the set of antiderivatives of the power function for p = -1. We have ... Referring to the table of differentials for the natural logarithm , Consequently, ... therefore .

I hope you grasp the principle.

Antiderivatives table ( indefinite integrals).

Formulas from the left column of the table are called basic antiderivatives. The formulas from the right column are not basic, but they are very often used to find indefinite integrals. They can be verified by differentiation.

Direct integration.

Direct integration is based on the use of the properties of indefinite integrals ,, integration rules and tables of antiderivatives.

Typically, the integrand must first be slightly transformed in order to use the table of basic integrals and the properties of the integrals.

Example.

Find the integral .

Solution.

Coefficient 3 can be taken out of the integral sign based on the property:

We transform the integrand (according to trigonometry formulas):

Since the integral of the sum is equal to the sum of the integrals, then

It's time to turn to the table of antiderivatives:

Answer:

.

Example.

Find the set of antiderivatives of a function

Solution.

Referring to the table of antiderivatives for the exponential function: ... I.e, .

Using the integration rule , then we have:

Thus, the table of antiderivatives, together with the properties and the rule of integration, allow us to find a lot of indefinite integrals. However, it is not always possible to transform the integrand to use the table of antiderivatives.

For example, in the table of antiderivatives, there is no integral of the logarithm function, arcsine function, arccosine, arctangent and arccotangent, tangent and cotangent function. To find them, special methods are used. But more on that in the next section: