Question

For the following exercises, integrate using whatever method you choose.\(\int x^{3} \sqrt{x^{2}+2} d x\)

Short answer

\( \int x^{3} \sqrt{x^{2}+2} \, dx = \frac{1}{5} (x^{2} + 2)^{5/2} - \frac{2}{3} (x^{2} + 2)^{3/2} + C \).

Step-by-step solution

Choose a Method

To solve the integral \( \int x^{3} \sqrt{x^{2}+2} \, dx \), we can use a substitution method. This is because the expression \( x^{2}+2 \) under the square root suggests a substitution that reduces complexity.

Substitution

Let \( u = x^{2} + 2 \). Then, differentiate \( u \) with respect to \( x \) to find \( du \):\[ du = 2x \, dx \]Thus, \( x \, dx = \frac{1}{2} \, du \).Rewrite the integral in terms of \( u \):\[ \int x^{3} \sqrt{x^{2}+2} \, dx = \int x^{2} \cdot x \sqrt{u} \cdot dx \]Since \( x^{2} = u - 2 \), substitute to get:\[ \int (u - 2) x \sqrt{u} \cdot dx = \int (u - 2) \sqrt{u} \cdot \frac{1}{2} \, du \]

Simplify and Integrate

Expand and simplify the integrand:\[ \int \frac{1}{2} ((u - 2) \cdot \sqrt{u}) \, du = \int \frac{1}{2} (u^{3/2} - 2u^{1/2}) \, du \]Separate the integral:\[ \frac{1}{2} \int u^{3/2} \, du - \int u^{1/2} \, du \]Integrate each term:\[ \frac{1}{2} \cdot \frac{2}{5} u^{5/2} - \frac{2}{3} u^{3/2} = \frac{1}{5} u^{5/2} - \frac{2}{3} u^{3/2} \]

Substitute Back

Replace \( u \) with \( x^{2} + 2 \):\[ \frac{1}{5} (x^{2} + 2)^{5/2} - \frac{2}{3} (x^{2} + 2)^{3/2} + C \]This gives the final solution to the integral in terms of \( x \).

Key concepts

Substitution Method

The substitution method is a powerful technique in integration that simplifies complex integrals by transforming them into easier ones. It involves changing the variable of integration to one that reduces the complexity of the integrand.
For the integral \( \int x^{3} \sqrt{x^{2}+2} \, dx \), it is beneficial to substitute the expression inside the square root.

• Start by letting \( u = x^{2} + 2 \). This choice is intuitive because it appears as an inner function within the composite function \( \sqrt{x^{2}+2} \).
• Differentiating \( u \) with respect to \( x \), we get \( du = 2x \, dx \).
• Solving for \( x \, dx \), we find \( x \, dx = \frac{1}{2} \, du \).

By substituting these values back into the integral, you transform a complicated expression into one that's easier to handle, making the integration process smoother and more straightforward.

Definite Integrals

Definite integrals refer to the computation of the integral between two specified limits. While the exercise did not involve specific limits, understanding definite integrals is crucial.
Definite integrals allow us to calculate the accumulated area under a curve defined by a function between two points.
They are represented as:
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \] where \( F \) is the antiderivative of \( f \).

• When using substitution in a definite integral, remember to adjust the limits of integration according to your substitution.
• If \( u = g(x) \), replace \( a \) and \( b \) with \( g(a) \) and \( g(b) \) respectively.

This ensures that the limits of integration correctly correspond to the new variable. Understanding this concept is essential for correctly solving problems involving definite integrals.

Integrands Simplification

Simplifying integrands is a critical step in solving integrals efficiently. This process involves breaking down the integrand to a form that is easier to integrate.
Consider the original integrand \( x^{3} \sqrt{x^{2}+2} \). Without simplification, this integral looks intimidating. Simplification involves a few systematic steps:

• Substitute \( u = x^{2} + 2 \) to transform the expression under the square root.
• Resolve and express \( x^{2} \) in terms of \( u \), yielding \( x^{2} = u - 2 \).
• Rewrite the integrand as \( (u - 2)x \sqrt{u} \).
• Use the relationship \( x \, dx = \frac{1}{2} \, du \) from the substitution to express everything in terms of \( u \).

These steps produce a new integrand \( \frac{1}{2}(u^{3/2} - 2u^{1/2}) \) that is simple to integrate, illustrating how simplification can make integration far more approachable.

Integration Step-by-Step Solution

Solving integrals using a step-by-step approach is beneficial for clarity and understanding.
This involves addressing small parts of the problem in a logical progression until you reach the solution. Here is how you can think about the provided solution:

• Step 1: Identify and choose a suitable method, like substitution, to simplify the task.
• Step 2: Perform the substitution properly, changing variables to reduce the complexity of the integrand.
• Step 3: Simplify the integrand and integrate each term separately.
• Step 4: Substitute back the original variable to express the antiderivative in terms of the initial variable.

By integrating each term, you can develop a final solution in terms of \( x \) like \( \frac{1}{5} (x^{2} + 2)^{5/2} - \frac{2}{3} (x^{2} + 2)^{3/2} + C \).
Adopting this systematic approach ensures accuracy and reinforces the foundational techniques of integration.