Question

Evaluate each definite integral to three significant digits. Check some by calculator. $$\int_{2}^{4}(x+3)^{2} d x$$

Short answer

\[57.3\]

Step-by-step solution

Expand the Integrand

Expand the quadratic expression \(x+3)^2\) using the binomial formula: \(x+3)^2 = x^2 + 2\cdot x\cdot3 + 3^2 = x^2 + 6x + 9\).

Set Up the Integral with Expanded Integrand

Replace \(x+3)^2\) with its expanded form in the integral: \[\int_{2}^{4}(x^2 + 6x + 9) dx\].

Find the Antiderivative

Calculate the antiderivative of \(x^2 + 6x + 9\), which is \frac{x^3}{3} + 3x^2 + 9x\.

Evaluate the Antiderivative from Lower to Upper Limit

Substitute the limits of the integral into the antiderivative and find the difference: \[\left(\frac{4^3}{3} + 3\cdot4^2 + 9\cdot4\right) - \left(\frac{2^3}{3} + 3\cdot2^2 + 9\cdot2\right)\].

Calculate and Simplify

Calculate the values: \[\frac{64}{3} + 48 + 36 - \left(\frac{8}{3} + 12 + 18\right)\] and simplify it to obtain the value of the definite integral.

Key concepts

Binomial Expansion

When faced with an expression like \((x+3)^2\), we can use the binomial expansion to rewrite it as a sum of terms. The binomial theorem tells us that any binomial raised to a power can be expanded into a sum involving terms of the form \(a^nb^m\), where \(n\) and \(m\) are non-negative integers that sum up to the power, and \(a\) and \(b\) are the base of the binomial. In this exercise, the expansion \((x+3)^2\) turns into \(x^2 + 6x + 9\), with \(x^2\) representing the term where we have two \(x\)'s multiplied (\(x*x\)), the term \(6x\) coming from two instances of \(3*x\), and the constant \(9\) being the result of \(3*3\). Expanding the binomial makes the process of integration straightforward as it transforms the integral of a polynomial into a sum of integrals of simpler functions.

Antiderivative Calculation

The antiderivative, also known as the indefinite integral, is the reverse process of differentiation. When we have a function like \(x^2 + 6x + 9\), finding its antiderivative means we're looking for a function which, when differentiated, gives us back our original function. To calculate the antiderivative step by step, we use the power rule, which says to add 1 to the exponent and divide by the new exponent. The antiderivative of \(x^2\) is \(\frac{x^3}{3}\), for \(6x\) it's \(3x^2\) (since \(\frac{6}{2}=3\) as the power of \(x\) increases by 1), and for \(9\) it's \(9x\). The complete antiderivative of our function is thus \(\frac{x^3}{3} + 3x^2 + 9x\). Recognizing the antiderivative is essential for evaluating definite integrals.

Integral Limits Substitution

Once we have calculated the antiderivative, the next step in evaluating a definite integral is to perform the limits substitution. This practically means plugging the upper and lower limits of the integral into the antiderivative we've found.

For the integral we're considering, our limits are \(2\) and \(4\). We substitute these values into our antiderivative, \(\frac{x^3}{3} + 3x^2 + 9x\), one by one. Evaluating it at \(4\) gives us \(\frac{4^3}{3} + 3\cdot4^2 + 9\cdot4\), and evaluating it at \(2\) gives us \(\frac{2^3}{3} + 3\cdot2^2 + 9\cdot2\). The next step is to find the difference between these two calculations to get the value of the definite integral.

Definite Integral Simplification

The final step in evaluating the definite integral is simplification. This involves carrying out the arithmetic in the expression resulting from the limits substitution. We take the difference between the value of the antiderivative evaluated at the upper limit and the lower limit. Since the upper limit gives us \(\frac{64}{3} + 48 + 36\) and the lower limit gives us \(\frac{8}{3} + 12 + 18\), subtracting these amounts yields \(\frac{64}{3} - \frac{8}{3}\) plus \(48 - 12\) plus \(36 - 18\). Simplify each separate term first, then combine them to find the value of the integral.

Remember, each step in the evaluation of a definite integral -- from expansion, through antiderivative calculation and limits substitution, to simplification -- builds upon the previous steps. It's important to take care and be precise at every stage to ensure a correct final answer.