Integral long logarithm formula derivation. What is a logarithm? Solving logarithms
Table of antiderivatives.
The properties of the indefinite integral allow one to find its antiderivative using a known differential of a function. Thus, using equalities and can be found from the table of derivatives of the main ones elementary functions make a table of antiderivatives.
Let us remind you table of derivatives, let's write it down in the form of differentials.
For example, let's find the indefinite integral power function.
Using the differential table , therefore, from the properties of the indefinite integral we have . That's why or in another post
Let's find the set of antiderivatives of the power function for p = -1. We have . We refer to the table of differentials for the natural logarithm , hence, . That's why .
I hope you understand the principle.
Table of antiderivatives (indefinite integrals).
The formulas from the left column of the table are called basic antiderivatives. The formulas in the right column are not basic, but are very often used when finding indefinite integrals. They can be checked 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 first needs to be slightly transformed so that the table of basic integrals and properties of integrals can be used.
Example.
Find the integral .
Solution.
Coefficient 3 can be removed from the integral sign based on the property:
Let's transform integrand function(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.
We refer to the table of antiderivatives for exponential function: . That is, .
If we use the integration rule , then we have:
Thus, the table of antiderivatives, together with the properties and the rule of integration, make it possible to find a lot of indefinite integrals. However, it is not always possible to transform the integrand function in order to use the table of antiderivatives.
For example, in the table of antiderivatives there is no integral of the logarithm function, arcsine, arccosine, arctangent and arccotangent, tangent and cotangent functions. To find them, use special methods. But more on that in the next section:
Table of antiderivatives ("integrals"). Table of integrals. Tabular not definite integrals. (The simplest integrals and integrals with a parameter). Formulas for integration by parts. Newton-Leibniz formula.
Table of antiderivatives ("integrals"). Tabular indefinite integrals. (The simplest integrals and integrals with a parameter). |
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Integral of a power function. |
Integral of a power function. |
An integral that reduces to the integral of a power function if x is driven under the differential sign. |
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Integral of an exponential, where a is a constant number. |
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Integral complex exponential function. |
Integral of an exponential function. |
An integral equal to the natural logarithm. |
Integral: "Long logarithm". |
Integral: "Long logarithm". |
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Integral: "High logarithm". |
An integral, where x in the numerator is placed under the differential sign (the constant under the sign can be either added or subtracted), is ultimately similar to an integral equal to the natural logarithm. |
Integral: "High logarithm". |
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Cosine integral. |
Sine integral. |
Integral equal to tangent. |
Integral equal to cotangent. |
Integral equal to both arcsine and arccosine |
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An integral equal to both arcsine and arccosine. |
An integral equal to both arctangent and arccotangent. |
Integral equal to cosecant. |
Integral equal to secant. |
Integral equal to arcsecant. |
Integral equal to arccosecant. |
Integral equal to arcsecant. |
Integral equal to arcsecant. |
Integral equal to the hyperbolic sine. |
Integral equal to 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. |
Formulas for integration by parts. Integration rules.
Formulas for integration by parts. Newton-Leibniz formula. Rules of integration. |
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Integrating a product (function) by a constant: |
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Integrating the sum of functions: |
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indefinite integrals: |
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Formula for integration by parts definite integrals: |
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Newton-Leibniz formula definite integrals: |
Where F(a),F(b) are the values of the antiderivatives at points b and a, respectively. |
Table of derivatives. Tabular derivatives. Derivative of the product. Derivative of the quotient. Derivative complex function.
If x is an independent variable, then:
Table of derivatives. Tabular derivatives."table derivative" - yes, unfortunately, this is exactly how they are searched for on the Internet |
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Derivative of a power function |
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Derivative of the exponent |
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Derivative of a complex exponential function |
Derivative of exponential function |
Derivative of a logarithmic function |
Derivative of the natural logarithm |
Derivative of the natural logarithm of a function |
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Derivative of sine |
Derivative of cosine |
Derivative of cosecant |
Derivative of a secant |
Derivative of arcsine |
Derivative of arc cosine |
Derivative of arcsine |
Derivative of arc cosine |
Tangent derivative |
Derivative of cotangent |
Derivative of arctangent |
Derivative of arc cotangent |
Derivative of arctangent |
Derivative of arc cotangent |
Derivative of arcsecant |
Derivative of arccosecant |
Derivative of arcsecant |
Derivative of arccosecant |
Derivative of the hyperbolic sine Derivative of the hyperbolic sine in the English version |
Derivative of hyperbolic cosine Derivative of hyperbolic cosine in English version |
Derivative of hyperbolic tangent |
Derivative of hyperbolic cotangent |
Derivative of the hyperbolic secant |
Derivative of the hyperbolic cosecant |
Rules of differentiation. Derivative of the product. Derivative of the quotient. Derivative of a complex function. |
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Derivative of a product (function) by a constant: |
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Derivative of sum (functions): |
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Derivative of the product (functions): |
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Derivative of the quotient (of functions): |
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Derivative of a complex function: |
Properties of logarithms. Basic formulas for logarithms. Decimal (lg) and natural logarithms (ln).
Basic logarithmic identity |
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Let's show how any function of the form a b can be made exponential. Since a function of the form e x is called exponential, then |
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Any function of the form a b can be represented as a power of ten |
Natural logarithm ln (logarithm to base e = 2.718281828459045...) ln(e)=1; ln(1)=0
Taylor series. Taylor series expansion of a function.
It turns out that the majority practically encountered mathematical functions can be represented with any accuracy in the vicinity of a certain point in the form of power series containing powers of a variable in increasing order. For example, in the vicinity of the point x=1:
When using series called Taylor's rows mixed functions containing, say, algebraic, trigonometric and exponential functions can be expressed as purely algebraic functions. Using series, you can often quickly perform differentiation and integration.
The Taylor series in the neighborhood of point a has the form:
1)
, where f(x) is a function that has derivatives of all orders at x=a. R n - the remainder term in 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 (the expansion occurs around the point a=0)
at a=0
members of the series are determined by the formula
Conditions for using Taylor series.
1. In order 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 term in the Taylor (Maclaurin (=McLaren)) formula for this function tends to zero as k →∞ on the specified interval (-R;R).
2. It is necessary that there are derivatives for a 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 in the domain of definition of f converges to f in some neighborhood of a.
There are infinitely differentiable functions whose Taylor series converges, but at the same time differs from the function in any neighborhood of a. For example:
Taylor series are used in approximation (approximation - scientific method, which consists in replacing some objects with others, in one sense or another close to the original ones, 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 the analysis of a linear system, in some sense equivalent to the original one.) equations occurs by expanding into a Taylor series and cutting off all terms above first order.
Thus, almost any function can be represented as a polynomial with a given accuracy.
Examples of some common expansions of power functions in Maclaurin series (=McLaren, Taylor in the vicinity of point 0) and Taylor in the vicinity of point 1. The first terms of expansions of the main functions in 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
What is a logarithm?
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
What is a logarithm? How to solve logarithms? These questions confuse many graduates. Traditionally, the topic of logarithms is considered complex, incomprehensible and scary. Especially equations with logarithms.
This is absolutely not true. Absolutely! Don't believe me? Fine. Now, in just 10 - 20 minutes you:
1. You will understand what is a logarithm.
2. Learn to decide whole class exponential equations. Even if you haven't heard anything about them.
3. Learn to calculate simple logarithms.
Moreover, for this you will only need to know the multiplication table and how to raise a number to a power...
I feel like you have doubts... Well, okay, mark the time! Let's go!
First, solve this equation in your head:
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You can get acquainted with functions and derivatives.