Question

In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals. $$ \int x^{2} \arctan x d x $$

Short answer

Choose \( u = \arctan x \) and \( dv = x^2 \, dx \).

Step-by-step solution

Identify the Structure for Integration by Parts

The formula for integration by parts is \( \int u \ dv = uv - \int v \ du \). This requires choosing functions \( u \) and \( dv \) such that the integration simplifies.

Define u and dv

For the integral \( \int x^2 \arctan x \, dx \), decide which part to be \( u \) and which to be \( dv \). Based on LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) rule, choose \( u = \arctan x \) (inverse trigonometric is a good choice for \( u \)) and hence \( dv = x^2 \, dx \).

Compute du and v

Differentiate \( u \) to find \( du \): \[ du = \frac{1}{1 + x^2} \, dx \] Now integrate \( dv \) to find \( v \): \[ v = \int x^2 \, dx = \frac{x^3}{3} \]

Set Up the Integration by Parts Formula

Substitute \( u \), \( du \), \( v \), and \( dv \) into the integration by parts formula: \[ \int x^2 \arctan x \, dx = \left( \arctan x \cdot \frac{x^3}{3} \right) - \int \left( \frac{x^3}{3} \cdot \frac{1}{1+x^2} \, dx \right) \] The formula is set up, and no evaluation is required as per instructions.

Key concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on finding integrals. The integral of a function represents the area under its curve. This process is inverse to differentiation. In differential calculus, we find rates of change, whereas in integral calculus, we compute the accumulation of quantities. The basic idea is to sum up infinitesimal parts to find the whole.

One of the key techniques in integral calculus is integration by parts. This technique is particularly useful when the integral is a product of two functions. It allows us to transform a more complicated integral into a simpler one by appropriately choosing parts of the integrand to differentiate and integrate.

• Integration by parts is derived from the product rule of differentiation.
• The formula is: \[ \int u \, dv = uv - \int v \, du \]
• Choosing a correct function for \( u \) can simplify the integral.

Remember, integration by parts is often used when dealing with products of polynomials and other functions such as inverse trigonometric or logarithmic functions.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse operations of the basic trigonometric functions like sine, cosine, and tangent. They are used to find angles when their trigonometric ratios are known. These functions are crucial in many applications such as physics and engineering.

The main inverse trigonometric functions include \( \arcsin(x), \arccos(x), \text{and} \ \arctan(x) \). Each has a specific range and domain to maintain a one-to-one correspondence with their respective trigonometric functions.

• For example, \( \arctan(x) \) is the inverse function of \( \tan(x) \), with a range of \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).
• These functions are essential for dealing with angles and solving certain integrals.

Inverse trigonometric functions often appear in integration problems, and they are a typical choice for the \( u \) function in integration by parts due to their properties when differentiated.

LIATE Rule

The LIATE rule is a useful mnemonic for choosing the function to be \( u \) in integration by parts. It stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. This rule provides a guideline on the preference for choosing \( u \), aiming to simplify the integration process.

Here is the priority within the LIATE rule:

• Logarithmic functions: Give highest preference (e.g., \( \ln x \)).
• Inverse trigonometric functions: Second preference (e.g., \( \arctan x \)).
• Algebraic functions: Followed next (e.g., \( x^2 \)).
• Trigonometric functions: Next in consideration (e.g., \( \sin x \)).
• Exponential functions: Lowest preference (e.g., \( e^x \)).

The choice of \( u \) according to LIATE helps not only simplify the integral but also ensures that the derivative \( du \) is simpler than \( u \). This rule guided the choice of \( \arctan x \) as \( u \) in our original problem, making the integration more manageable.