Evaluate a definite integral by using a substitution to simplify a square root, expanding the integrand, and applying the power rule.

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Evaluating a Definite Integral with a Square Root and a Polynomial

The definite integral core

Evaluate the integral: \[ \int_{-1}^2 \sqrt{x+2} x^2 \, dx \]

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What you need first

Integration by substitution
Power rule for integration
Evaluating definite integrals
Fractional exponents
Arithmetic with fractions

What this problem tests

Tests integration by substitution, changing limits of integration, expanding polynomials with fractional exponents, and careful arithmetic with fractions.

This is a standard problem in a first-year calculus course, testing the ability to perform integration by substitution and handle the resulting algebraic expansion and arithmetic.

Evaluating a Definite Integral with a Square Root and a Polynomial

What you need first

What this problem tests

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