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Integration by Parts Calculator + Online Solver With Free Steps

Integration by Parts is an online tool that offers an antiderivative or represents the area under a curve. This method reduces the integrals to standard forms from which the integrals can be determined.

This Integration by Parts calculator uses all feasible ways for the integration and offers solutions with stages for each. Given that users may enter different math operations using the keyboard, its usability is excellent.

The Integration by Parts Calculator is capable of integrating functions with numerous variables as well as definite and indefinite integrals (antiderivatives).

What Is an Integration by Parts Calculator?

Integration by Parts Calculator is a calculator that uses a calculus approach for determining the integral of a functioning product in terms of the integrals of its derivative and antiderivative.

In essence, the integration by parts formula changes the antiderivative of the functions into a different form so that it is simpler to discover the simplify/solve if you have an equation with the antiderivative of two functions multiplied together and don’t know how to calculate the antiderivative.

Here is the formula:

\[\int_{}^{}(u\cdot v)dx = u\int_{}^{}(v)dx −\int_{}^{}\frac{du}{dx}[\int_{}^{}(v)dx]dx\]

The antiderivative of the product of two functions, which is where you begin, is transformed to the right side of the equation.

If you need to determine the antiderivative of a complex function that is challenging to solve without splitting it into two functions multiplied together, you can utilize integration by parts.

How To Use an Integration by Parts Calculator?

You can use the Integration by Parts Calculator by following the given guidelines, and the calculator will then provide you with the desired results. You can follow the given instructions below to get the solution of Integral for the given equation.

Step 1

Choose your variables.

Step 2

Differentiate u in relevance to x to find $\frac{du}{dx}$

Step 3

Integrate v to find $\int_{}^{}v dx$

Step 4

To solve for integration by parts, enter these values.

Step 5

Click on the “SUBMIT” button to get the integral solution and also the whole step-by-step solution for the Integration by Parts will be displayed.

Finally, in the new window, the graph of the area under the curve will be displayed.

How Does Integration by Parts Calculator Work?

Integration by Parts Calculator works by moving the product out of the equation so that the integral can be evaluated easily and it replaces a difficult integral with one that is easier to evaluate.

Finding the integral of the product of two distinct types of functions, such as logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential functions, is done using the integration by parts formula.

The integral of a product can be calculated using the integration by parts formula u . v, U(x), and V(x) can be chosen in any order when applying the product rule of differentiation to differentiate a product.

However, when utilizing the integration by parts formula, we must first determine which of the following functions appears first in the following order before assuming it is the first function, u(x).

  • Logarithmic (L)
  • Inverse trigonometric (I)
  • Algebraic (A)
  • Trigonometric (T)
  • Exponential (E)

The ILATE rule is used to keep this in mind. For instance, if we need to determine the value of x ln x dx (x is a certain algebraic function while ln is a logarithmic function), we will place ln x to be u(x) since, in LIATE, the logarithmic function comes first. There are two definitions for the integration by parts formula. Either of them can be used to integrate the result of two functions.

What Is Integration?

Integration is a method that solves the differential equation of path integrals. The area under a graph’s curve is calculated using integral function differentiation.

Integrand in Integration Calculator

The integrand is represented by function f, which is an integral equation or integration formula (x). You must input the value in the integration calculator for it to function properly.

How Does the Integral Calculator Deal With Integral Notation?

The calculator deals with integral notation by calculating its integral using laws of integration.

For an integral equation:

\[\int_{}^{}(2x) \cdot dx\]

$\int_{}^{}$ is the Integral Symbol and 2x is the function we want to integrate.

The differential of the variable x in this integral equation is denoted by dx. It indicates that the variable in the Integration is x. The dx and dy symbols indicate the orientation along the x- and y-axes, respectively.

The integrals calculator uses the integral sign and integral rules to produce results rapidly.

Integration by Parts Formula Derivation

The formula for the derivative of the product of two functions can be used to prove integration by parts. The derivative of the product of the two functions f(x) and g(x) is equal to the product of the derivatives of the first function multiplied by the second function and its derivative multiplied by the first function for the two functions f(x) and g(x).

Let’s use the product rule of differentiation to derive the integration by parts equation. Take u and v, two functions. Let y i.e., y = u . v, be their output. By utilizing the principle of product differentiation, we obtain:

\[\frac{d}{dx} (u \cdot v) = u (\frac{dv}{dx} + v (\frac{du}{dx})\]

We will rearrange the terms here.

\[u (\frac{dv}{dx}) = \frac{d}{dx} (u \cdot v)  – v (\frac{du}{dx})\]

Integrating on both sides with respect to x:

\[\int_{}^{}u (\frac{dv}{dx}) (dx) = \int_{}^{} \frac{d}{dx} (u \cdot v)   dx – \int_{}^{}v (\frac{du}{dx}) dx\]

By cancelling the terms:

\[\int_{}^{}u dv = uv – \int_{}^{}v du\]

Thus, the formula for integration by parts is derived.

Functions and integrals can both be evaluated with the use of an integral calculator by parts. The tool helps us save time that would otherwise be spent performing calculations manually.

Additionally, it aids in providing the integration result without charge. It works swiftly and gives immediate, accurate results.

This online calculator offers results that are clear and step-by-step. This online calculator can be used to solve equations or functions involving definite or indefinite integrals.

Formulas Related to Integration by Parts

The following formulas, which are useful when integrating different algebraic equations, were derived from the integration by parts formula.

\[\int_{}^{} e^x (f(x) + f'(x)) \cdot dx = e^x \cdot f(x) + C \]

\[\int_{}^{} \sqrt{(x^2 + a^2)} \cdot dx = \frac{1}{2} \cdot x \cdot \sqrt (x^2 + a^2)+ \frac{a^2}{2} \cdot log|x + \sqrt{(x^2 + a^2)}| +C \]

Benefits of Using Integration by Parts Calculator

The benefits of using this Integration by Parts Calculator are:

  1. The integral by parts calculator makes it possible to calculate the integration by parts using both definite and indefinite integrals.
  2. The calculator eliminates the need for manual computations or drawn-out processes by quickly solving integral equations or functions.
  3. The online tool saves time and gives the solution to many equations in a short amount of time.
  4. This calculator will enable you to practice consolidating your integration by parts principles and will show you the results step-by-step.
  5. You will receive a plot and any potential intermediate steps of integration by parts from this calculator.
  6. The results of this online calculator will include the real component, imaginary part, and alternative form of the integrals.

Solved Examples

Let’s look at some detailed examples to better understand the concept of the Integration by Parts Calculator.

Example 1

Solve \[\int_{}^{}x \cdot \cos(x) dx\] by using integration by parts method.

Solution

Given that:

\[\int_{}^{}x \cdot \cos(x) dx\]

The formula of integration by parts is \[\int_{}^{}(u.v)dx = u\int_{}^{}(v)dx -\int_{}^{}\frac{du}{dx}[\int_{}^{}(v)dx]dx\]

So, u=x

du=dx

dv= cos(x)

\[\int_{}^{}\cos(x) dx= \sin(x)\]

By substituting the values in the formula:

\[\int_{}^{}x\cdot \cos(x) dx= x\cdot \sin(x)-\int_{}^{}\sin(x) dx\]

=x.sin(x) + cos(x)

Therefore, \[\int_{}^{}x \cdot \cos(x) dx=x\cdot \sin(x)+\cos(x)+C\]

Example 2

Find \[\int_{}^{}x \cdot \sin(x) dx\]

Solution

Given that:

u= x

\[\frac{du}{dx}= 1\]

v=sin(x)

\[\int_{}^{}v\ dx=\int_{}^{}\sin(x)\ dx=-\cos (x)\]

Now it’s time to insert the variables into the formula:

\[\int_{}^{}(u.v)dx = u\int_{}^{}(v)dx -\int_{}^{}\frac{du}{dx}[\int_{}^{}(v)dx]dx\]

This will give us:

\[\int_{}^{}(x.sin(x))dx = x\int_{}^{}(\sin x)dx -\int_{}^{}\frac{d(x)}{dx}[\int_{}^{}(\sin x)dx]\]

\[\int_{}^{}(x\cdot \sin(x))dx = x(-\cos x) -\int_{}^{}1.[\int_{}^{}(\sin x)dx]\]

\[\int_{}^{}(x\cdot \sin(x))dx = x(-\cos x) -1.\int_{}^{}(-\cos x)dx\]

Next, we will work the right side of the equation to simplify it. First distribute the negatives:

\[\int_{}^{}(x\cdot \sin(x))dx = x(-\cos x) +1.\sin x\]

The integrations of cos x is sin x, and make sure to add the arbitrary constant, C, at the end:

\[\int_{}^{}(x\cdot \sin(x))dx = -x(\cos x) +\sin x+C\]

That’s it, you found the Integral!

Example 3

Find \[\int_{}^{}x^2 \cdot \ln{x}dx\]

Solution

Given that,

u= ln(x)

\[\frac{du}{dx}= \frac{1}{x}\]

\[v=x^2\]

\[\int_{}^{}v\ dx=\int_{}^{}x^2\ dx=\frac{x^3}{3}\]

Now that we know all the variables, let’s plug them into the equation:

\[\int_{}^{}(u\cdot v)dx = u\int_{}^{}(v)dx – \int_{}^{}\frac{du}{dx}[\int_{}^{}(v)dx]dx\]

\[\int_{}^{}(x^2 \cdot \ln{x})dx = \ln{x}\cdot \frac{x^3}{3} – \int_{}^{}\frac{1}{x}[\frac{x^3}{3}]dx\]

The last thing to do now is to simplify! First, multiply everything out:

\[\int_{}^{}(x^2 \cdot \ln{x})dx = \ln{x} \cdot \frac{x^3}{3} -\int_{}^{}\frac{x^2}{3}dx\]

\[\int_{}^{}(x^2 \cdot \ln{x})dx = \frac{x^3 \cdot \ln{x}}{3} -\frac{x^3}{9}+C\]

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