Integration by Part Calculator

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. The calculator on this page provides a quick and accurate way to compute integrals using the integration by parts formula.

What is Integration by Parts?

Integration by parts is a technique derived from the product rule for differentiation. The product rule states that if u and v are functions of x, then:

d/dx (u·v) = u·dv/dx + v·du/dx

Rearranging this equation gives us the integration by parts formula:

∫u·dv = u·v - ∫v·du

This formula allows us to convert a difficult integral into an easier one. The choice of u and dv is crucial and often requires some trial and error to find the most effective pair.

How to Use the Calculator

Using the integration by parts calculator is straightforward. Follow these steps:

Enter the function you want to integrate in the "Function" field.
Select the variable of integration (usually x).
Choose the functions u and dv that you believe will simplify the integral.
Click the "Calculate" button to compute the integral.
Review the result and the step-by-step solution provided.

The calculator will display the result of the integration by parts process, including the final integral value and a breakdown of each step.

Integration by Parts Formula

The integration by parts formula is given by:

∫u·dv = u·v - ∫v·du

Where:

u is a function that is differentiated to simplify the integral.
dv is a function that is integrated.
du is the derivative of u with respect to x.
v is the antiderivative of dv with respect to x.

The choice of u and dv is critical. A common strategy is to choose u as the function that becomes simpler when differentiated, and dv as the function that can be easily integrated.

Step-by-Step Example

Let's solve the integral ∫x·e^x dx using integration by parts.

Choose u = x and dv = e^x dx.
Compute du = dx and v = e^x.
Apply the integration by parts formula: ∫x·e^x dx = x·e^x - ∫e^x dx.
Compute ∫e^x dx = e^x + C.
Combine the results: ∫x·e^x dx = x·e^x - e^x + C.

The final result is:

∫x·e^x dx = (x - 1)e^x + C

Common Integrals Solved with Integration by Parts

Integration by parts is commonly used to solve integrals involving products of polynomials and exponential, trigonometric, or logarithmic functions. Some examples include:

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

These examples demonstrate the versatility of integration by parts in solving a wide range of integrals.

Frequently Asked Questions

What is the integration by parts formula?

The integration by parts formula is ∫u·dv = u·v - ∫v·du, where u and dv are functions chosen to simplify the integral.

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

Choose u as the function that becomes simpler when differentiated, and dv as the function that can be easily integrated. Common strategies include using the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential).

Can integration by parts be used for all integrals?

No, integration by parts is most effective for integrals involving products of functions. It may not be the best method for simple integrals that can be solved using basic rules.

What is the LIATE rule?

The LIATE rule is a mnemonic used to choose u and dv in integration by parts. It stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions, indicating the order of preference for choosing u.