Integration by Parts Calculator with Steps

Integration by parts is a fundamental technique in calculus used to find integrals of products of functions. This calculator provides step-by-step solutions for integrals of the form ∫u dv, helping students and professionals solve complex integration problems efficiently.

What is Integration by Parts?

Integration by parts is a method based on the product rule for differentiation. It's particularly useful when dealing with integrals of products of functions, where one function is easy to differentiate and the other is easy to integrate.

The formula for integration by parts is derived from the product rule:

If d/dx(u) = u' and d/dx(v) = v', then:

d/dx(uv) = u'v + uv'

Integrating both sides with respect to x gives:

uv = ∫u'v dx + ∫uv' dx

Rearranged, this becomes the integration by parts formula:

∫u dv = uv - ∫v du

This technique is especially valuable when dealing with integrals involving logarithmic, inverse trigonometric, or exponential functions multiplied by polynomials.

How to Use the Calculator

Enter the function you want to integrate in the "Function u" field.
Enter the function you want to differentiate in the "Function dv" field.
Click the "Calculate" button to see the step-by-step solution.
Review the result and the detailed steps shown below the calculator.

For best results, enter functions in terms of x. The calculator handles basic algebraic functions and some transcendental functions.

Integration by Parts Formula

The integration by parts formula is:

∫u dv = uv - ∫v du

Where:

u is a differentiable function of x
dv is a differential of another function
v is the antiderivative of dv
du is the differential of u

The formula works by transforming the original integral into a simpler form that can be integrated more easily.

Worked Example

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

Choose u = x and dv = e^x dx
Then du = dx and v = e^x
Apply the formula: ∫x e^x dx = x e^x - ∫e^x dx
Integrate the remaining term: ∫e^x dx = e^x + C
Combine results: ∫x e^x dx = x e^x - e^x + C = e^x (x - 1) + C

This example shows how integration by parts can simplify complex integrals into more manageable forms.

Common Mistakes

When using integration by parts, several common errors can occur:

Choosing u and dv incorrectly, leading to more complex integrals
Forgetting to include the constant of integration (C)
Miscounting the number of integration by parts steps needed
Making sign errors when applying the formula

To avoid these mistakes, carefully select u and dv based on the functions' differentiability and integrability, and double-check each step of the process.

FAQ

When should I use integration by parts?

Use integration by parts when you have an integral of the form ∫u dv, where u is a function that can be differentiated and dv is a differential that can be integrated.

How do I choose u and dv?

Choose u to be the function that becomes simpler when differentiated, and dv to be the function that can be easily integrated. Common choices include polynomials for u and logarithmic or inverse trigonometric functions for dv.

What if I need to use integration by parts multiple times?

If the remaining integral after one application of integration by parts is still complex, you may need to apply the method again. Continue until the integral becomes simple enough to solve.