Integral Calculator by Parts

Integration by parts is a fundamental technique in calculus used to find the integral of products of functions. This method is particularly useful when dealing with integrals that cannot be solved using basic integration rules. Our integral calculator by parts provides a step-by-step solution to help you understand and solve complex integrals efficiently.

What is Integration by Parts?

Integration by parts is a technique derived from the product rule for differentiation. It allows us to integrate products of functions by expressing them in terms of simpler integrals. The method is based on the formula:

Integration by Parts Formula

∫u dv = uv - ∫v du

Where:

u and v are functions of x
du and dv are their derivatives

The formula works by "parts" - we choose one function to be u and the other to be dv, then differentiate and integrate accordingly. The choice of u and dv can significantly affect the complexity of the resulting integral.

When to Use Integration by Parts

Integration by parts is particularly useful when:

The integrand is a product of two functions
One function is easily integrable
The other function's derivative simplifies the integral

Integration by Parts Formula

The integration by parts formula is derived from the product rule of differentiation. The product rule states that if you have two functions u(x) and v(x), then:

Product Rule

d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

Rearranging this equation to solve for the integral of u(x)v(x) gives us the integration by parts formula:

Integration by Parts Formula

∫u dv = uv - ∫v du

This formula allows us to transform the integral of a product into a simpler expression involving the product of u and v minus the integral of v times du.

Choosing u and dv

The choice of u and dv is crucial in integration by parts. A common strategy is the LIATE rule:

Logarithmic functions
Inverse trigonometric functions
Algebraic functions
Trigonometric functions
Exponential functions

According to the LIATE rule, you should choose u to be the function that appears highest in this list. This often simplifies the integral.

How to Use the Calculator

Our integral calculator by parts is designed to make solving integrals using integration by parts as simple as possible. Here's how to use it:

Enter the function you want to integrate in the "Function" field
Select the variable of integration (usually x)
Choose the lower and upper limits of integration
Select which part of the function to use as u (u or dv)
Click "Calculate" to see the step-by-step solution

The calculator will display:

The integral you entered
The chosen u and dv
The derivatives and integrals used
The final result
A graphical representation of the function and its integral

Calculator Limitations

While our calculator provides accurate results for most integrals, it may not be able to solve all types of integrals. For complex integrals, you may need to use more advanced techniques or symbolic computation software.

Worked Example

Let's solve the integral ∫x e^x dx using integration by parts. We'll use the calculator to verify our steps.

Step 1: Choose u and dv

According to the LIATE rule, we choose u = x (algebraic) and dv = e^x dx (exponential).

Step 2: Find du and v

Differentiate u to find du:

du = d/dx [x] dx = 1 dx

Integrate dv to find v:

v = ∫e^x dx = e^x

Step 3: Apply the integration by parts formula

∫x e^x dx = uv - ∫v du = x e^x - ∫e^x (1 dx)

Step 4: Solve the remaining integral

∫e^x dx = e^x + C

Step 5: Combine results

∫x e^x dx = x e^x - e^x + C = e^x (x - 1) + C

Verification

Using our calculator with u = x and dv = e^x dx, we get the same result: e^x (x - 1) + C.

Common Integrals Solved by Parts

Many integrals that are difficult to solve using basic techniques can be solved using integration by parts. Here are some common examples:

Integral	Solution
∫x e^x dx	e^x (x - 1) + C
∫x^2 e^x dx	e^x (x^2 - 2x + 2) + C
∫ln x dx	x ln x - x + C
∫x sin x dx	sin x - x cos x + C
∫x cos x dx	cos x + x sin x + C

These examples demonstrate how integration by parts can transform complex integrals into simpler forms. Our calculator can solve these and many other integrals efficiently.

Frequently Asked Questions

What is the integration by parts formula?

The integration by parts formula is ∫u dv = uv - ∫v du, where u and dv are functions of x, and du and v are their derivatives and integrals respectively.

When should I use integration by parts?

You should use integration by parts when dealing with integrals of products of functions, especially when one function is easily integrable and the derivative of the other simplifies the integral.

How do I choose u and dv in integration by parts?

A common strategy is to use the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to choose u as the function that appears highest in this list.

Can integration by parts solve all integrals?

No, integration by parts is not a universal method. It works best for integrals of products of functions and may not be applicable to all types of integrals.

What if I can't solve the remaining integral after using integration by parts?

If the remaining integral is still complex, you may need to apply integration by parts again or use other techniques such as substitution or partial fractions.