How to Integrate x^2 + x^2 = 1: Solve the Equation

x^2 + x^2 = 1 is an equation, not a function to integrate directly. Combining the like terms gives 2x^2 = 1, so x^2 = 1/2 and x = ±1/√2 ≈ ±0.7071. That means how to integrate x 2 x 2 1 is really two tasks in one: first solve the equation, then integrate the resulting quadratic form. Once you reduce it to 2x^2 = 1, the integration is a standard power rule and produces a general antiderivative involving x^3/3.

Solve x^2 + x^2 = 1

Start by combining like terms. Both terms are the same monomial, x^2, so x^2 + x^2 = 2x^2. The equation is now:

2x^2 = 1

Divide both sides by 2:

x^2 = 1/2

Take the square root of both sides. Because squaring a positive or negative number gives the same result, you get two real solutions:

x = ±1/√2 = ±√2/2 ≈ ±0.7071

This “plus or minus” step is the part people most often forget. You can verify the answer by substituting back: 2(1/√2)^2 = 2(1/2) = 1, which checks out. The simplification and square-root method are standard algebra covered in introductory courses such as Khan Academy.

Integrate the Resulting Expression

To integrate, first isolate the quadratic expression: 2x^2 = 1, so x^2 = 1/2. Integrating x^2 uses the power rule for integrals:

∫x^2 dx = x^3/3 + C

where C is the constant of integration — any constant, because its derivative is zero. Multiplying by 2 to match the original left side gives the general antiderivative of 2x^2:

∫2x^2 dx = 2x^3/3 + C

Quick worked example: integrate 2x^2 from x = 0 to x = 1. The antiderivative is 2x^3/3, so evaluating gives (2·1^3/3) − (2·0^3/3) = 2/3 ≈ 0.6667. This is a definite integral, so no constant of integration remains. Both the power rule and the constant of integration are documented in standard references like Wolfram MathWorld.

One key distinction: solving the equation yields specific x-values, while integrating yields a family of functions (indefinite) or a single number (definite). Keep those two goals separate, and the problem becomes straightforward.

Key Takeaways

• x^2 + x^2 = 1 simplifies to 2x^2 = 1, giving x = ±1/√2 ≈ ±0.7071.
• To integrate, use the power rule: ∫x^2 dx = x^3/3 + C, so ∫2x^2 dx = 2x^3/3 + C.
• Always include the constant of integration C for indefinite integrals; omit it for definite integrals.
• Verify solutions by substituting back — squaring can hide a sign error, so check both roots.