Question

Simplify and integrate. $$\int \sqrt{x}(3 x-2) d x$$

Short answer

\(\frac{6}{5}x^{\frac{5}{2}} - \frac{4}{3}x^{\frac{3}{2}} + C\)

Step-by-step solution

Simplify the integrand

Rewrite the integrand by distributing the square root over each term in the parentheses. This means you will multiply \(\sqrt{x}\) by each term inside the parentheses. The integrand becomes \(3x^{\frac{3}{2}} - 2x^{\frac{1}{2}}\).

Set up the integral with simplified integrand

Now write the integral with the simplified integrand: \[\int (3x^{\frac{3}{2}} - 2x^{\frac{1}{2}}) dx\].

Integrate term by term

Integrate each term separately using the power rule, which means you add 1 to the exponent and divide by the new exponent. The integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\), so for the first term: \(\int 3x^{\frac{3}{2}} dx = 3 \int x^{\frac{3}{2}} dx = 3 \frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}\) and for the second term: \(\int -2x^{\frac{1}{2}} dx = -2 \int x^{\frac{1}{2}} dx = -2 \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}\).

Simplify the integrated terms

Simplify each term from the previous step: For the first term \(3 \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = \frac{3 \times 2}{5}x^{\frac{5}{2}} = \frac{6}{5}x^{\frac{5}{2}}\) and for the second term \( -2 \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = -\frac{2 \times 2}{3}x^{\frac{3}{2}} = -\frac{4}{3}x^{\frac{3}{2}}\).

Write the complete integral with the constant of integration

Combine the simplified integrated terms and add the constant of integration, C: \[\frac{6}{5}x^{\frac{5}{2}} - \frac{4}{3}x^{\frac{3}{2}} + C\].

Key concepts

Indefinite Integral

The indefinite integral represents the general form of the antiderivative of a function. It's used to reverse the process of differentiation, much like solving a puzzle by working backwards. In essence, if you're given the rate of change of a function (its derivative), the indefinite integral helps you find the original function, often referred to as the primitive or antiderivative.

Every time we perform an indefinite integration, we include a 'constant of integration' to account for any constant term that was lost during differentiation. This constant represents all the possible vertical translations of the antiderivative. Integrating a function like \( \sqrt{x}(3x - 2) dx\) involves undoing the differentiation, and by doing so, we recover the form of the original function plus that constant.

Power Rule for Integration

The power rule for integration is a basic and powerful tool in calculus, which states that to integrate a term of the form \(x^n\), you add 1 to the exponent and then divide by the new exponent, thus obtaining \(\frac{x^{n+1}}{n+1}\).

However, it's important to note that this rule only applies when \(n eq -1\), since that would lead to the natural logarithm function instead of the power function. In the context of square roots, the square root of x, or \(\sqrt{x}\), can be expressed as \(x^{\frac{1}{2}}\), which allows us to apply the power rule directly after rewriting the square root in exponential form. For example, integrating \(\sqrt{x}\) itself would result in \(\frac{2}{3}x^{\frac{3}{2}}\), since \(\frac{1}{2} + 1 = \frac{3}{2}\) and then we divide by \(\frac{3}{2}\).

Simplifying Integrands

Simplifying integrands is often a crucial first step in performing integration. It involves rewriting the expression under the integral sign into a form that makes it easier to apply integration rules. This can include distributing multiplication over addition, factoring, expanding, and using trigonometric identities, among others.

In the case of \(\sqrt{x}(3x - 2) dx\), the integrand is simplified by distributing the \(\sqrt{x}\) over each term within the parentheses. This results in an expression with exponents of a half, which are more straightforward to integrate using the power rule. Thus, the original complex expression is decomposed into simpler terms that are separately integrated. Such simplifications can significantly reduce the complexity of the integral and help in avoiding common mistakes.

Constant of Integration

The constant of integration, typically denoted by \(C\), is a crucial component of the indefinite integral. It represents an unknown constant that could have been part of the original function before differentiation occurred.

Mathematically, because differentiation of a constant results in zero, the integration process cannot reveal the exact value of this constant solely based on the derivative. Therefore, every indefinite integral will include this constant of integration to encompass the family of antiderivatives that are possible. When integrating a function term by term, you only need to add the constant of integration once at the end of the process, as combining multiple antiderivatives from a sum does not affect the value of this unknown constant.