Integration by Completing The Square Calculator

Integration by completing the square is a technique used to evaluate integrals of the form ∫x²e^(ax² + bx + c)dx. This method transforms the integrand into a perfect square, making it easier to integrate. Our calculator performs this transformation automatically and provides step-by-step results.

What is Integration by Completing the Square?

Integration by completing the square is a method used to evaluate integrals of the form:

∫x²e^(ax² + bx + c)dx

The technique involves rewriting the quadratic expression inside the exponent as a perfect square plus a constant. This transformation simplifies the integral into a form that can be integrated using standard techniques.

The general approach involves:

Identifying the quadratic expression in the exponent
Completing the square to rewrite it as (x + d)² + e
Using substitution to simplify the integral
Integrating the transformed expression

How to Use the Method

Step 1: Identify the Quadratic Expression

First, identify the quadratic expression in the exponent. For example, in ∫x²e^(2x² + 4x + 3)dx, the quadratic is 2x² + 4x + 3.

Step 2: Complete the Square

Rewrite the quadratic expression as a perfect square plus a constant. For 2x² + 4x + 3:

Factor out the coefficient of x²: 2(x² + 2x) + 3
Complete the square inside the parentheses: x² + 2x + 1 - 1 = (x + 1)² - 1
Multiply back by 2: 2[(x + 1)² - 1] + 3 = 2(x + 1)² - 2 + 3 = 2(x + 1)² + 1

Step 3: Rewrite the Integral

Substitute the completed square back into the integral:

∫x²e^(2(x + 1)² + 1)dx = e^1 ∫x²e^(2(x + 1)²)dx

Step 4: Use Substitution

Let u = x + 1, then du = dx. The integral becomes:

e ∫(u - 1)²e^(2u²)du

Step 5: Expand and Integrate

Expand (u - 1)² to u² - 2u + 1 and integrate term by term:

e [∫u²e^(2u²)du - 2∫ue^(2u²)du + ∫e^(2u²)du]

Each of these integrals can be evaluated using standard techniques.

Worked Examples

Example 1: Basic Integration

Evaluate ∫x²e^(x² + 2x + 1)dx

Complete the square: x² + 2x + 1 = (x + 1)²
Rewrite the integral: ∫x²e^((x + 1)²)dx
Let u = x + 1, du = dx: ∫(u - 1)²e^(u²)du
Expand: ∫(u² - 2u + 1)e^(u²)du
Integrate term by term: (1/2)e^(u²) - ue^(u²) + (1/2)∫e^(u²)du
Substitute back: (1/2)e^((x + 1)²) - (x + 1)e^((x + 1)²) + (1/2)∫e^(u²)du + C

Example 2: With Coefficient

Evaluate ∫x²e^(2x² + 4x + 3)dx

Complete the square: 2x² + 4x + 3 = 2(x² + 2x + 1) + 1 = 2(x + 1)² + 1
Rewrite the integral: e^1 ∫x²e^(2(x + 1)²)dx
Let u = x + 1, du = dx: e ∫(u - 1)²e^(2u²)du
Expand: e [∫(u² - 2u + 1)e^(2u²)du]
Integrate term by term: e [(1/4)e^(2u²) - (1/2)ue^(2u²) + (1/4)∫e^(2u²)du]
Substitute back: (e/4)e^(2(x + 1)²) - (e/2)(x + 1)e^(2(x + 1)²) + (e/4)∫e^(2u²)du + C

Applications

Integration by completing the square is particularly useful in:

Evaluating integrals of the form ∫x²e^(ax² + bx + c)dx
Solving differential equations involving exponential functions
Physics problems involving heat conduction and diffusion
Engineering applications involving exponential decay or growth

Comparison of Integration Methods
Method	Applicable Integrals	Advantages	Limitations
Completing the Square	∫x²e^(ax² + bx + c)dx	Works for quadratic exponents	Requires careful algebraic manipulation
Integration by Parts	∫x e^x dx	Versatile for many functions	Can be time-consuming
Substitution	∫e^(x²)dx	Simple when substitution is obvious	Requires recognizing substitution

FAQ

When should I use integration by completing the square?: Use this method when you have an integral of the form ∫x²e^(ax² + bx + c)dx. It's particularly effective when the exponent contains a quadratic expression.
What if the quadratic doesn't complete to a perfect square?: If the quadratic doesn't complete to a perfect square, you may need to use other integration techniques or consider numerical methods for approximation.
Can this method be used for higher powers of x?: This method is specifically designed for integrals involving x². For higher powers, different techniques may be more appropriate.
Is there a way to automate this process?: Yes, our calculator automates the process of completing the square and performing the integration, providing step-by-step results.
What if the integral doesn't converge?: If the integral doesn't converge, the method will still attempt to find a solution, but the result may involve special functions or be undefined.