Problem 33 - Calculus Single Variable

Question

Integrate by parts to evaluate the given definite integral. $$ \int_{0}^{\sqrt{3}} \arcsin (x / 2) d x $$

Step-by-Step Solution

Answer

\( \sqrt{3}\pi/3 - 1 \).

Step 1: Identify Parts for Integration by Parts

In the integration by parts formula, \( \int u \, dv = uv - \int v \, du \), you need to choose \( u \) and \( dv \). Here, let \( u = \arcsin(x/2) \) and \( dv = dx \).

Step 2: Differentiate and Integrate Parts

First, differentiate \( u \) to find \( du \). The derivative of \( \arcsin(x/2) \) with respect to \( x \) is \( du = \frac{1}{\sqrt{1 - (x/2)^2}} \cdot \frac{1}{2} \, dx = \frac{1}{\sqrt{4 - x^2}} \, dx \). Integrate \( dv = dx \) to get \( v = x \).

Step 3: Apply the Integration by Parts Formula

Substitute \( u \), \( v \), and \( du \) into the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). This becomes: \[ \int \arcsin(x/2) \, dx = x \arcsin(x/2) - \int \frac{x}{\sqrt{4 - x^2}} \, dx \].

Step 4: Simplify and Evaluate Remaining Integral

The integral \( \int \frac{x}{\sqrt{4 - x^2}} \, dx \) can be solved using substitution. Let \( t = \sqrt{4 - x^2} \), then \( x = \sqrt{4 - t^2} \) and \( dx = \frac{-t}{\sqrt{4 - t^2}} \, dt \). Simplifying gives: \[ \int \frac{x}{\sqrt{4 - x^2}} \, dx = \int -dt = -t \]. Since \( t = \sqrt{4 - x^2} \), this becomes \( -\sqrt{4-x^2} \).

Step 5: Substitute Back and Evaluate Definite Integral

Combine results from Steps 3 and 4: \[ \int \arcsin(x/2) \, dx = x \arcsin(x/2) + \sqrt{4-x^2} \]. Evaluate from \( x = 0 \) to \( x = \sqrt{3} \). Compute: \[ \left[ x \arcsin(x/2) + \sqrt{4-x^2} \right]_0^{\sqrt{3}} \].

Step 6: Final Evaluation

At \( x = \sqrt{3} \), \( \arcsin(\sqrt{3}/2) = \pi/3 \). Substitute into \( x \arcsin(x/2) + \sqrt{4-x^2} \): \[ \sqrt{3} (\pi/3) + \sqrt{4-3} = \sqrt{3}\pi/3 + 1 \]. At \( x = 0 \), \( \arcsin(0) = 0 \) and \( \sqrt{4-0} = 2 \). Compute: \[ (\sqrt{3}\pi/3 + 1) - (0 + 2) = \sqrt{3}\pi/3 - 1 \].

Key Concepts

Definite Integral

A definite integral is used to find the area under a curve within certain boundaries. Imagine slicing a shape into thin strips, adding their areas together gives the total area. In mathematics, a definite integral, written as \( \int_{a}^{b} f(x) \, dx \), represents this concept. Here, \( a \) and \( b \) are the limits of integration, marking the beginning and end of the region over which the function \( f(x) \) is integrated.

Evaluating a definite integral involves two key steps:

• First, find the antiderivative (indefinite integral) of the function.
• Second, calculate the difference between the antiderivative's values at the upper and lower limits. This step uses the Fundamental Theorem of Calculus.

In the context of our exercise, we calculate \( \int_{0}^{\sqrt{3}} \arcsin(x/2) \, dx \), meaning we find the area under the graph of \( \arcsin(x/2) \) from \( x = 0 \) to \( x = \sqrt{3} \). This process gives us a specific numerical value indicating the accumulated area within these boundaries.

Arcsine Function

The arcsine function is the inverse of the sine function, meaning it gives you the angle whose sine is a given number. If \( y = \sin(x) \), then \( x = \arcsin(y) \). This function is important in trigonometry as it helps in solving problems that involve right angled triangles when their sides are known.

In our problem, \( \arcsin(x/2) \) is the expression we are integrating. The function has a restricted domain and range:

• Domain: \([-1, 1]\), because sine only returns values within this interval.
• Range: \([-\pi/2, \pi/2]\), as these are the limits where arcsine is typically defined.

When you integrate a function involving arcsine, as in the exercise, it often requires careful handling due to its inverse nature. Differentiating \( \arcsin(x/2) \) gives \( \frac{1}{\sqrt{4-x^2}} \), which is necessary for integration by parts, a method that simplifies integrating products of functions.

Substitution Method

The substitution method in calculus is a technique used to simplify the process of finding an antiderivative, especially when dealing with complex expressions. It involves changing variable names or changing a part of the function for easier integration.

Here’s how it works:

• Select a substitution to simplify the function, usually setting \( u \) to a part of the function that makes differentiation simpler.
• Rewrite the entire integral in terms of the new variable \( u \).
• Perform the integration with respect to \( u \).
• Substitute back the original variable.

In the step by step solution, substitution is used to tackle the integral \( \int \frac{x}{\sqrt{4-x^2}} \, dx \). By setting \( t = \sqrt{4-x^2} \), the integration becomes manageable. This substitution helps transform the problem into a simpler form, where the resulting integral can be evaluated directly. Once it's solved, we substitute back the original terms, aiding in completing the definite integration process.