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Example
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1
Solved example of calculus
$\int\cos\left(3x\right)\cdotd\cdot x\cdot dx$
2
We can solve the integral $\int\cos\left(3x\right)dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let’s call it $u$), which when substituted makes the integral easier. We see that $3x$ it’s a good candidate for substitution. Let’s define a variable $u$ and assign it to the choosen part
$u=3x$
Intermediate steps
Differentiate both sides of the equation $u=3x$
$du=\frac{d}{dx}\left(3x\right)$
Find the derivative
$\frac{d}{dx}\left(3x\right)$
The derivative of the linear function times a constant, is equal to the constant
$3$
3
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
$du=3dx$
4
Isolate $dx$ in the previous equation
$\frac{du}{3}=dx$
5
Substituting $u$ and $dx$ in the integral and simplify
$\int\frac{\cos\left(u\right)}{3}du$
Intermediate steps
Take the constant $\frac{1}{3}$ out of the integral
$\frac{1}{3}\int\cos\left(u\right)du$
Divide $1$ by $3$
$\frac{1}{3}\int\cos\left(u\right)du$
6
Take the constant $\frac{1}{3}$ out of the integral
$\frac{1}{3}\int\cos\left(u\right)du$
7
Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$
$\frac{1}{3}\sin\left(u\right)$
Intermediate steps
$\frac{1}{3}\sin\left(3x\right)$
8
Replace $u$ with the value that we assigned to it in the beginning: $3x$
$\frac{1}{3}\sin\left(3x\right)$
9
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$\frac{1}{3}\sin\left(3x\right)+C_0$
Final Answer
$\frac{1}{3}\sin\left(3x\right)+C_0$
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Derivative Calculator: Wolfram|Alpha
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What are derivatives?
The derivative is an important tool in calculus that represents an infinitesimal change in a function with respect to one of its variables.
Given a function , there are many ways to denote the derivative of with respect to . The most common ways are and . When a derivative is taken times, the notation or is used. These are called higher-order derivatives. Note for second-order derivatives, the notation is often used.
At a point , the derivative is defined to be . This limit is not guaranteed to exist, but if it does, is said to be differentiable at . Geometrically speaking, is the slope of the tangent line of at .
As an example, if , then and then we can compute : .
The derivative is a powerful tool with many applications. For example, it is used to find local/global extrema, find inflection points, solve optimization problems and describe the motion of objects.
How Wolfram|Alpha calculates derivatives
Wolfram|Alpha calls Wolfram Languages’s D function, which uses a table of identities much larger than one would find in a standard calculus textbook. It uses well-known rules such as the linearity of the derivative, product rule, power rule, chain rule and so on. Additionally, D uses lesser-known rules to calculate the derivative of a wide array of special functions. For higher-order derivatives, certain rules, like the general Leibniz product rule, can speed up calculations.
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Integrals Calculator • Step by step!
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Did my calculator help you? Tell your friends about this Calculator and you can help me too!
At the top of the page, enter the function you want to integrate.
Integration variable , integration limits and other parameters can be changed in section “ Settings “. Press “ = ” to start integration/antiderivative. The result will be shown below on this page.
How the Integral Calculator works
For those who are interested in the technical details, this section describes how the Integrals Calculator works and works.
First, the parser analyzes the original mathematical expression. It converts it into a form more convenient for the computer, namely into the form of a tree (see the picture below). In the process of such transformation, the Integral Calculator must respect the order of operations, taking into account their priority. As well as the fact that in mathematical expressions the multiplication sign is often omitted, for example, we usually write “5x” instead of “5*x”. An integral calculator should be able to understand such cases and add the multiplication sign itself.
The parser is written in JavaScript and is based on the marshalling yard algorithm, so it can be executed directly in the browser.
This makes it possible to generate a human-readable expression on the fly, converting the resulting tree into code for LaTeX (LaTeX). With the help of MathJax, the image is generated and displayed in the browser.
On pressing the “=” button, the Integral Calculator sends the mathematical expression along with the parameters (integration variable and integration limits) to the server, where it is analyzed again. This time, the expression is converted into a form that will be understood by the computer algebra system Maxima (Maxima).
Maxima calculates the integral of a mathematical function. Maxima’s result is again converted to LaTeX and then shown to the user. The antiderivative is calculated using the Risch algorithm, which is quite sophisticated for human understanding. That is why the task of showing intermediate steps for solving integrals is so difficult.
In order to still show a step-by-step solution, the Integral Calculator uses the same methods that a person would use.
The algorithm that does this was developed over several years and was written in Maxima’s own programming language. The program contains more than 17,000 lines of code. If the expression to be integrated has the same form as a known one, the algorithm applies predefined rules for solving the integral (for example, the method of indefinite coefficients for rational functions, trigonometric substitution in square root integrals of a quadratic function, or integration by parts for products of certain functions). If it does not coincide with the already known, then the algorithm tries different substitutions and transformations until the integral is solved or until the time allotted for this ends, or until all possible options run out. On the one hand, the Calculator does not have mathematical intuition, which would be of great help in the search for an antiderivative, but, on the other hand, the Calculator is able to try a large number of different options in a very short time. Such a step-by-step calculation of the antiderivative according to the rules is often more compact and elegant than that calculated by Maxima.
Another “Check Solution” mode of operation is to solve the complex problem of determining whether two mathematical expressions are equal to each other. The difference between the expressions is calculated and simplified using the Maxima as much as possible. For example, it could be rewriting trigonometric/hyperbolic functions into their exponential forms. If it is possible to simplify the difference to zero, the task is completed. Otherwise, a probabilistic algorithm is applied that evaluates and compares both expressions at random locations. In the case of the antiderivative, the whole procedure is repeated for each derivative, since the antiderivative may differ by a constant.
Interactive function graphs are computed in the browser and rendered on canvas from HTML5. For each math function that needs to be drawn, the Calculator creates a JavaScript function, which is then calculated in increments necessary to display the graph correctly. All singularities (e.g. poles) of a function are detected during rendering and handled separately.