The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (dθ / √(2θ - θ²))

Verified step by step guidance

1. Start by examining the integral: \(\int \frac{d\theta}{\sqrt{2\theta - \theta^{2}}}\). Notice that the expression under the square root is a quadratic in \(\theta\).
2. Rewrite the quadratic expression inside the square root in a more recognizable form by completing the square: \(2\theta - \theta^{2} = -\left(\theta^{2} - 2\theta\right) = -\left(\theta^{2} - 2\theta + 1 - 1\right) = -\left( (\theta - 1)^{2} - 1 \right) = 1 - (\theta - 1)^{2}\).
3. Substitute \(x = \theta - 1\) to simplify the integral. Then, \(d\theta = dx\), and the integral becomes \(\int \frac{dx}{\sqrt{1 - x^{2}}}\).
4. Recognize that the integral \(\int \frac{dx}{\sqrt{1 - x^{2}}}\) is a standard form whose antiderivative is \(\arcsin x + C\).
5. Finally, substitute back \(x = \theta - 1\) to express the answer in terms of the original variable \(\theta\): the integral equals \(\arcsin(\theta - 1) + C\).

Key Concepts

Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. By letting a new variable represent a part of the integrand, the integral can often be transformed into a more manageable form. This technique is especially useful when the integrand contains composite functions.

Completing the Square

Completing the square rewrites a quadratic expression in the form ax² + bx + c as a perfect square plus or minus a constant. This method helps simplify expressions under square roots, making them easier to integrate, often leading to trigonometric substitutions.

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving square roots of quadratic polynomials with trigonometric functions. This leverages identities like sin²x + cos²x = 1 to simplify integrals, especially those involving √(a² - x²), √(x² - a²), or √(x² + a²).