Problem 36

Question

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{x}{\sqrt{2+3 x}} d x $$

Step-by-Step Solution

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Answer

The indefinite integral, or antiderivative, of $ \frac{x}{\sqrt{2+3 x}} $ is $ \frac{2}{3}\sqrt{2+3x}+\frac{4}{3(2+3x)^{1/2}} + C $.

1Step 1: Define the Substitution

Define $u$ to be the function inside the square root, $u = 2 + 3x$. Taking the derivative of $u$, we get $du = 3dx$. Divide by 3 to isolate $dx$, so $dx = \frac{1}{3} du$.

2Step 2: Replace the Variables in the Integral

Replace $2 + 3x$ with $u$, and $dx$ with $\frac{1}{3} du$. Then, the original integral $\int \frac{x}{\sqrt{2+3 x}} dx$ becomes $\frac{1}{3}\int \frac{u-2}{\sqrt{u}} du$.

3Step 3: Simplify the Integral and Solve

Simplify the integral into two separate fractions: $\frac{1}{3}\int[u^{-1/2} - 2u^{-3/2}] du$. Now, you can solve the integral: $ \frac{1}{3} [2u^{1/2} + 4u^{-1/2}]+ C$.

4Step 4: Substitute $u$ back in

Finally, replace $u$ with $2 + 3x$, to get the answer in terms of $x$: $ \frac{2}{3}\sqrt{2+3x}+\frac{4}{3(2+3x)^{1/2}} + C $.

Key Concepts

Indefinite integralSubstitution methodIntegral simplification

Indefinite integral

An indefinite integral is a fundamental concept in calculus, referring to the antiderivative of a given function. It represents a family of functions whose derivative is the original function we started with. The indefinite integral is shown using the integral symbol $ \int $, followed by a function and then $ dx $, indicating integration with respect to $ x $. It does not have specified limits which means it represents all possible antiderivatives with an added constant $ C $, called the constant of integration.
For example, if we are given the function $ f(x) = x^2 $, its indefinite integral is $ \int x^2 \, dx = \frac{x^3}{3} + C $. This is because when you take the derivative of $ \frac{x^3}{3} + C $, you get back $ x^2 $.
When computing indefinite integrals, it's important to identify any potential functions that could use techniques like substitution to simplify the process.

Substitution method

The substitution method is a powerful tool for solving complex integrals by transforming them into a simpler form. This technique involves choosing a part of the integral to substitute with a single variable, often denoted $ u $. The goal is to make the integral easier to evaluate by changing the variables.
Here's how you use substitution in three straightforward steps:

Select $ u $: Choose a term inside the integral, usually something inside a product or a composite function. For instance, if you have $ \int \frac{x}{\sqrt{2+3x}} \,dx $, selecting $ u = 2+3x $ simplifies the expression.
Differentiate and isolate $ dx $: Derive $ u $ to get $ du $ and replace $ dx $ in terms of $ du $ for a calculus transformation.
Substitute and integrate: Insert $ u $ into the integral, converting it into something you can solve, such as replacing $ \frac{x}{\sqrt{2+3x}} \,dx $ with $ \frac{1}{3} \int \frac{u-2}{\sqrt{u}} \,du $.

By following these steps, the integral is reformulated into a form that is often simpler to solve.

Integral simplification

Once the substitution method is applied, the next step often involves integral simplification. This process helps break down the integral into forms that are readily integrable. Simplification may involve distributing terms, breaking down fractions, or using standard integral forms to identify the solution.
Consider an example where after substitution you end up with an integral like $ \frac{1}{3}\int[u^{-1/2} - 2u^{-3/2}] \,du $. Here, simplification involves recognizing standard integral forms and solving:

$ \int u^{-1/2} \,du = 2u^{1/2} $, because the power rule for integration states that integrating $ u^n $ results in $ \frac{u^{n+1}}{n+1} $.
$ \int u^{-3/2} \,du = -\frac{2}{u^{1/2}} $, which follows the same principle.

Once simplified, the integral is then easy to solve, resulting in an expression that can be converted back to the original variable to provide the complete solution.

Problem 36

Problem 37

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Problem 36

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Problem 37

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let $n=4 .$.) $$ \in

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