Evaluate a definite integral involving a square root by using u-substitution and the Fundamental Theorem of Calculus.

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Definite Integral with a Square Root and Substitution

The definite integral core

\[ \int_{-1}^1 \sqrt{3x^2-2x+3} (3x-1) \, dx \]

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What you need first

Basic differentiation of polynomials
Fractional exponents
Simplifying square roots

What this problem tests

Tests the ability to recognize when to use u-substitution, correctly perform the substitution, integrate using the power rule, and evaluate the definite integral at its bounds.

This is a standard problem in a first-year Calculus course, typically encountered when learning integration techniques.

Definite Integral with a Square Root and Substitution

What you need first

What this problem tests

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