Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ arccos(x / 2) dx

Verified step by step guidance

1. Identify the integral to solve: \(\int \arccos\left(\frac{x}{2}\right) \, dx\).
2. Choose parts for integration by parts: let \(u = \arccos\left(\frac{x}{2}\right)\) and \(dv = dx\).
3. Compute \(du\): differentiate \(u\) with respect to \(x\). Use the chain rule on \(\arccos\left(\frac{x}{2}\right)\), recalling that \(\frac{d}{dx} \arccos(x) = -\frac{1}{\sqrt{1 - x^2}}\). So, \(du = -\frac{1}{\sqrt{1 - \left(\frac{x}{2}\right)^2}} \cdot \frac{1}{2} \, dx = -\frac{1}{2 \sqrt{1 - \frac{x^2}{4}}} \, dx\).
4. Compute \(v\): integrate \(dv = dx\), so \(v = x\).
5. Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Substitute the expressions for \(u\), \(v\), and \(du\) to set up the new integral to solve.

Key Concepts

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is crucial to simplify the integral effectively.

Inverse Trigonometric Functions

Inverse trigonometric functions, like arccos(x), are the inverses of trigonometric functions and have specific derivatives. For arccos(x), the derivative is -1/√(1 - x²). Understanding these derivatives helps in differentiating parts of the integrand during integration by parts.

Substitution and Simplification

Substitution involves changing variables to simplify integrals, especially when dealing with composite functions like arccos(x/2). Simplifying the resulting expressions after applying integration by parts often requires algebraic manipulation or trigonometric identities to reach a solvable form.