Instruction

1

Integration is an operation which is the opposite of differentiation. So, if you want to learn how to integrate, you first need to learn to find from any of the functions of derivatives. You can learn it quickly enough. After all, there is a special table of derivatives. It is already possible to solve simple integrals. And there are the table of basic indefinite integrals. It is presented in the figure.

2

Now we need to remember the most basic properties of integrals, given below.

3

Integral of sum of functions is best decomposed into a sum of integrals. This rule most often applies when the components function is fairly simple, if you can find them using the table of integrals.

4

There is one very important method. According to this method the function must be paid under the differential. They are especially good to use if before you make a differential, of a function take the derivative. Then it is put instead of dx. In that way we get df(x). This way you can easily ensure that even function under the differential can be used as an ordinary variable.

5

Another basic formula, without which very often just not do - this is the formula of integration by parts: Integral(udv)=uv-Integral(vdu). This formula is effective in the case if in the task it is required to find the integral of the product of two elementary functions. Of course you can use regular conversion, but it is difficult and time consuming. Therefore, to take the integral with the help of this formula is much simpler.

Useful advice

To solve the integral means to integrate in the variable set function. If the standard integral, we can say that it is almost solved. If he has a more complicated entry is main task when finding the integral of a function becomes bringing it to tabular form.

# Advice 2: How to solve derivatives

Differentiation (finding the derivative of the function) is the most important task of mathematical analysis. Finding the derivative of the function helps to explore the properties of functions, build its graph. Differentiation is used to solve many problems of physics and mathematics. How to learn how to take

**derivatives**?You will need

- Table of derivatives, notebook, pen

Instruction

1

Learn the definition of derivative. In principle, to take the derivative without knowing the definition of the derivative, but the understanding of what is happening in this case is negligible.

2

Make a table of derivatives, which record the

**derivatives**of basic elementary functions. Learn them. In any case, keep a table of derivatives is always at hand.3

See if you can simplify the function. In some cases, it's much easier to take the derivative.

4

The derivative of a constant function (constant) is zero.

5

From the definition of the derivative rules of differentiation are derived (the rules for finding the derivative). Learn those rules.Derivative of sum of functions is equal to the sum of the derivatives of these functions. The derivative of a difference of functions is equal to the difference between the derivatives of these functions. The sum and difference can be grouped under a single concept of algebraic sum.The constant factor can be taken out of the sign of the derivative.Derivative of product of two functions is equal to the sum of the products of the derivative of the first function and the second derivative of the second function on the first.Private derivative of two functions is equal to: the derivative of the first function times the second function minus the derivative of the second function times the first function, and to divide it by the square of the second function.

6

To take the derivative of a composite function, it is necessary to consistently present it in the form of elementary functions and take the derivative rules. It should be understood that one function can be an argument of another function.

7

Let us consider the geometric meaning of derivative. The derivative of the function at x is the tangent of the angle of inclination of the tangent to the graph of the function at the point X.

8

Practice. Start with finding the derivative of simple functions, and then move to more complex.

Useful advice

Self-print the basic rules of differentiation definition of a derivative. So you better learn and remember material.

# Advice 3: How to solve double integrals

From the course of mathematical analysis known concept of double integral. Geometrically, the double integral represents the volume of the cylindrical body based on D and bounded by the surface z = f(x, y). Using double integrals to calculate the mass of a thin plate with a given density, the area of a plane figure, the area of the piece of surface, the coordinates of the center of gravity of a homogeneous plate and the other quantities.

Instruction

1

The solution of the double integrals can be reduced to the computation of definite integrals.

If the function f(x, y) is a closed and continuous in a region D bounded by the line y = c and the line x = d, with c < d and a function y = g(x) and y = z(x), with g(x), z(x) is continuous on [c; d] and g(x) ? z(x) on this interval, then calculate the double integral by the formula presented in the figure.

If the function f(x, y) is a closed and continuous in a region D bounded by the line y = c and the line x = d, with c < d and a function y = g(x) and y = z(x), with g(x), z(x) is continuous on [c; d] and g(x) ? z(x) on this interval, then calculate the double integral by the formula presented in the figure.

2

If the function f(x, y) is a closed and continuous in a region D bounded by the line y = c and the line x = d, with c < d and a function y = g(x) and y = z(x), with g(x), z(x) is continuous on [c; d] and g(x) = z(x) on this interval, then calculate the double integral by the formula presented in the figure.

3

If you want to calculate the double integral more complex regions D, the region D is split into parts, each of which represents an area described in paragraph 1 or 2. Calculates the integral for each of these areas, the results obtained are summarized.

# Advice 4: How to solve derivatives

The derivative is one of the most important concepts not only in mathematics but also in many other fields of knowledge. It describes the rate of change of a function at a given point in time. From the point of view of geometry, the derivative at a point is the tangent of the angle of inclination of the tangent to that point. The process of finding is called differentiation, and backward - integration. Knowing a few simple rules, you can calculate derivative of any function, which in turn makes life easier and chemists, and physicists, and even microbiologists.

You will need

- a textbook of algebra for 9 grade.

Instruction

1

The first thing you need to differentiate the functions is to know the basic table of derivatives. It can be found in any mathematical Handbook.

2

In order to solve the problems associated with finding the derivative, you need to learn the basic rules. So, let's say we have two differentiable functions u and v and some constant value C.

Then:

Derivative of constants is always equal to zero: ()' = 0;

The constant is always imposed for the sign of the derivative: (cu)' = cu';

When finding the derivative of sum of two functions, you just need them to differentiate, and the results folded: (u+v)' = u'+v';

When finding the derivative of a product of two functions, you need the derivative of the first function times the second function and add the derivative of the second function multiplied by the first function: (u*v)' = u'*v+v'*u;

In order to find the derivative from a private two functions is necessary, from the product of the derivative of the dividend, multiplied by a function of the divisor, subtract the product of the derivative of the divisor, multiplied by a function of the dividend is divided by divisor function squared. (u/v)' = (u'*v-v'*u)/v^2;

If given a complex function, then multiply the derivative of the inner function and the derivative from the outside. Let y=u(v(x)), then y'(x)=y'(u)*v'(x).

Then:

Derivative of constants is always equal to zero: ()' = 0;

The constant is always imposed for the sign of the derivative: (cu)' = cu';

When finding the derivative of sum of two functions, you just need them to differentiate, and the results folded: (u+v)' = u'+v';

When finding the derivative of a product of two functions, you need the derivative of the first function times the second function and add the derivative of the second function multiplied by the first function: (u*v)' = u'*v+v'*u;

In order to find the derivative from a private two functions is necessary, from the product of the derivative of the dividend, multiplied by a function of the divisor, subtract the product of the derivative of the divisor, multiplied by a function of the dividend is divided by divisor function squared. (u/v)' = (u'*v-v'*u)/v^2;

If given a complex function, then multiply the derivative of the inner function and the derivative from the outside. Let y=u(v(x)), then y'(x)=y'(u)*v'(x).

3

Using the above-obtained knowledge, you can differentiate almost any function. So, let's look at some examples:

y=x^4, y'=4*x^(4-1)=4*x^3;

y=2*x^3*(e^x-x^2+6), y'=2*(3*x^2*(e^x-x^2+6)+x^3*(e^x-2*x));

Also there are challenges to calculating the derivative at a point. Imagine you are given the function y=e^(x^2+6x+5), you need to find the value of the function at x=1.

1) Find the derivative function: y'=e^(x^2-6x+5)*(2*x +6).

2) Calculate the value of the function at a given point y'(1)=8*e^0=8

y=x^4, y'=4*x^(4-1)=4*x^3;

y=2*x^3*(e^x-x^2+6), y'=2*(3*x^2*(e^x-x^2+6)+x^3*(e^x-2*x));

Also there are challenges to calculating the derivative at a point. Imagine you are given the function y=e^(x^2+6x+5), you need to find the value of the function at x=1.

1) Find the derivative function: y'=e^(x^2-6x+5)*(2*x +6).

2) Calculate the value of the function at a given point y'(1)=8*e^0=8

Useful advice

Learn the table of elementary derivatives. This will significantly save time.

# Advice 5: What the integrals

The integral is called the reciprocal of the differential of a function. Many physical and other problems are reduced to solving complex differential or integral equations. This requires knowing the differential and integral calculus.

Instruction

1

Imagine some function F(x), the derivative of which is f(x). This expression can be written in the following form:

F'(x)=f(x).

If the function f(x) is the derivative of the function F(x), the function F(x) is a primitive for f(x).

The same function can be somewhat primitive. An example of this might be a function x^2. It has an infinite number of primitives, among them major - such as x^3/3 or x^3/3+1. Instead of the units or any other number specified constant C, which is written as follows:

F(x)=x^n+C, where C=const.

The integration is called finding the integral of the function, inverse of the differential. The integral denoted by the sign ∫. It can be as vague, when given some function with an arbitrary C, and particular when it has some value. In this case, the integral is given by two values, called upper and lower limits.

F'(x)=f(x).

If the function f(x) is the derivative of the function F(x), the function F(x) is a primitive for f(x).

The same function can be somewhat primitive. An example of this might be a function x^2. It has an infinite number of primitives, among them major - such as x^3/3 or x^3/3+1. Instead of the units or any other number specified constant C, which is written as follows:

F(x)=x^n+C, where C=const.

The integration is called finding the integral of the function, inverse of the differential. The integral denoted by the sign ∫. It can be as vague, when given some function with an arbitrary C, and particular when it has some value. In this case, the integral is given by two values, called upper and lower limits.

2

Since the integral is an inverse value of the derivative in a General form it looks like this:

∫f(x)=F(x)+C.

For example, using a table of differentials, it is possible to find the integral of the function y=cosx:

∫cosx=sinx as the derivative of the function f(x) equal to f'(x)=(sinx)'=cosx.

The integrals there are other properties. Listed below are only the most basic of them:

- integral of the sum is equal to the sum of the integrals;

- the constant factor can be removed from the integral sign;

∫f(x)=F(x)+C.

For example, using a table of differentials, it is possible to find the integral of the function y=cosx:

∫cosx=sinx as the derivative of the function f(x) equal to f'(x)=(sinx)'=cosx.

The integrals there are other properties. Listed below are only the most basic of them:

- integral of the sum is equal to the sum of the integrals;

- the constant factor can be removed from the integral sign;

3

In some problems, especially in geometry and physics, applied integrals of another type - specific. For example, it can be used if necessary to determine the distance, which was a material point between time periods t1 and t2.

4

There are technical devices capable of integration. The simplest of them - integrating analog chain. It is available in integrating voltmeters, and in some of the dosimeters. Later was invented the digital integrators - pulse counters. Currently, the function of integrator you can assign software to any device that has a microprocessor.