Integration by Parts Calculator is a free online tool that displays the integrated value for the given function. STUDYQUERIES’S online integration by parts calculator tool makes the calculation faster, and it displays the integrated value in a fraction of seconds.

How to Use Integration by Parts Calculator?

The procedure to use the integration by parts calculator is as follows:

  • Step 1: Enter the function in the input field
  • Step 2: Now click the button “Evaluate the Integral” to get the output
  • Step 3: Finally, the integrated value will be displayed in the output field

Integration By Parts Calculator

What is Integration by Parts and Calculator?

Antiderivatives can be difficult enough to solve on their own, but when you’ve got two functions multiplied together that you need to take the antiderivative of, it can be difficult to know where to start. That’s where the integration by parts formula comes in!

Integration By Parts Calculator
Integration By Parts Calculator

This handy formula can make your calculus homework much easier by helping you find antiderivatives that otherwise would be difficult and time-consuming to work out. In this guide, we’ll explain the formula, walk you through each step you need to take to integrate by parts, and solve example problems so you can become an integration by parts expert yourself.

What Is the Integration by Parts Formula?

Integration by parts is a technique used in calculus to find the integral of a product of functions in terms of the integral of their derivative and antiderivative. Basically, if you have an equation with the antiderivative two functions multiplied together, and you don’t know how to find the antiderivative, the integration by parts formula transforms the antiderivative of the functions into a different form so that it’s easier to find the simplify/solve. Here is the formula:

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

You start with the left side of the equation (the antiderivative of the product of two functions) and transform it to the right side of the equation.

You can use integration by parts when you have to find the antiderivative of a complicated function that is difficult to solve without breaking it down into two functions multiplied together. It may not seem like an incredibly useful formula at first, since neither side of the equation is significantly more simplified than the other, but as we work through examples, you’ll see how useful the integration by parts formula can be for solving antiderivatives.

How to Solve Problems Using Integration by Parts

There are five steps to solving a problem using the integration by parts formula:

  1. Choose your u and v
  2. Differentiate u with respect to x to Find \(\pmb{\color{red}{\frac{du}{dx}}}\)
  3. Integrate v to find \(\pmb{\color{red}{\int_{}^{}v dx}}\)
  4. Plug these values into the integration by parts equation
  5. Simplify and solve

It may seem complicated to integrate by parts, but using the formula is actually pretty straightforward. The first three steps all have to do with choosing/finding the different variables so that they can be plugged into the equation in step four. You’ll have to have a solid grasp of how to differentiate and integrate, but if you do, those steps are easy.

Read Also: Inverse Property: Definition, Uses & Examples

In general, your goal is for du to be simpler than u and for the antiderivative of dv to not be any more complicated than v. Basically, you want the right side of the equation to stay as simple as possible to make it easier for you to simplify and solve. However, don’t stress too much over choosing your u and v. If your first choices don’t work, just switch them and integrate by parts with your new u and v to see if that works better.

Once you have your variables, all you have to do is simplify until you no longer have any antiderivatives, and you’ve got your answer! Keep reading to see how we use these steps to solve actual sample problems.

Integration by Parts Examples

\(\pmb{\color{red}{Find \int_{}^{}x.sin(x) dx}}\)

If you were to just look at this problem, you might have no idea how to go about taking the antiderivative of \(x.sin(x)\), so let’s start there.

\(u= x\)

\(\frac{du}{dx}= 1\)

\(v=sin(x)\)

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

Now it’s time to plug those variables into the integration by parts formula:

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

This gives us:

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

$$\int_{}^{}(x.sin(x))dx = x(-cosx) −\int_{}^{}1.[\int_{}^{}(sinx)dx]$$

$$\int_{}^{}(x.sin(x))dx = x(-cosx) -1.\int_{}^{}(-cosx)dx$$

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

$$\int_{}^{}(x.sin(x))dx = x(-cosx) +1.sinx$$

The integrations of cos(x) is sin(x), and don’t forget to add the arbitrary constant, C, at the end:

$$\int_{}^{}(x.sin(x))dx = -x(cosx) +sinx+C$$

That’s it, you found the Integral!

\(\pmb{\color{red}{Find \int_{}^{}x^2. ln(x) dx}}\)

\(u= ln(x)\)

\(\frac{du}{dx}= \frac{1}{x}\)

\(v=x^2\)

\(\int_{}^{}v\ dx=\int_{}^{}x^2\ dx=\frac{x^3}{3}\)

Now that we have all the variables, let’s plug them into the integration by parts equation:

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

$$\int_{}^{}(x^2. ln(x))dx = ln(x).\frac{x^3}{3} −\int_{}^{}\frac{1}{x}[\frac{x^3}{3}]dx$$

All that’s left now is to simplify! First multiply everything out:

$$\int_{}^{}(x^2. ln(x))dx = ln(x).\frac{x^3}{3} −\int_{}^{}\frac{x^2}{3}dx$$

$$\int_{}^{}(x^2. ln(x))dx = \frac{x^3.ln(x)}{3} −\frac{x^3}{9}+C$$

Conclusion

The integration by parts formula can be a great way to find the antiderivative of the product of two functions you otherwise wouldn’t know how to take the antiderivative of. You’ll need to have a solid knowledge of derivatives and antiderivatives to be able to use it, but it’s a straightforward formula that can help you solve various math problems. The steps are:

  1. Choose your u and v
  2. Differentiate u with respect to x to Find \(\pmb{\color{red}{\frac{du}{dx}}}\)
  3. Integrate v to find \(\pmb{\color{red}{\int_{}^{}v dx}}\)
  4. Plug these values into the integration by parts equation
  5. Simplify and solve

FAQs

What are integration and its formula?

Integration can be considered the reverse process of differentiation or called Inverse Differentiation. Integration is the process of finding a function with its derivative. Basic integration formulas on different functions are mentioned here.

What are the 5 basic integration formulas?

\(\int_{}^{}x^n.dx = \frac{x^{n+1}}{n+1} + C\)

\(\int_{}^{}cosx.dx = sin x + C.\)

\(\int_{}^{}sinx.dx = -cos x + C.\)

\(\int_{}^{}sec^2 x.dx = tan x + C.\)

\(\int_{}^{}cosec^2 x.dx = -cot x + C.\)

What is the integration of \(e^2x\)?

The integration of \(e^2x\) is \(\frac{e^{2x}}{2} + C\), by using the substitution method for the integration.

What is the integration of 1?

It is x+c. The differentiation of x with respect to x is 1. And, Integration is the reverse process of differentiation. So, integration of 1 is x+c, where c is the Constant of Integration.

What is the UV rule of integration?

UV integration is one of the important methods to solve integration problems. This method of integration is often used for integrating products of two functions. UV rule of integration: Let u and v are two functions then the formula of integration is. $$\int_{}^{}(u.v)dx = u\int_{}^{}(v)dx −\int_{}^{}\frac{du}{dx}[\int_{}^{}(v)dx]dx$$