Evaluate the following integrals.
65. ∫ from 0 to 1/6 1/√(1 - 9x²) dx

Verified step by step guidance

1. Step 1: Recognize the integral's structure. The integrand is of the form $1/\surd (1\; -\; u²)$, which resembles the derivative of the arcsine function. Specifically, the derivative of $arcsin(u)$ is $1/\surd (1\; -\; u²)$.
2. Step 2: Identify the substitution needed to simplify the integral. Let $u\; =\; 3x$, which implies $du\; =\; 3\; dx$. Rewrite the limits of integration accordingly: when $x\; =\; 0$, $u\; =\; 0$, and when $x\; =\; 1/6$, $u\; =\; 1/2$.
3. Step 3: Substitute into the integral. The integral becomes $\int \; from\; 0\; to\; 1/2\; (1/3)\; *\; (1/\surd (1\; -\; u²))\; du$, where the factor $1/3$ comes from adjusting for $du\; =\; 3\; dx$.
4. Step 4: Apply the antiderivative of $1/\surd (1\; -\; u²)$, which is $arcsin(u)$. The integral becomes $(1/3)\; *\; [arcsin(u)]$, evaluated from $u\; =\; 0$ to $u\; =\; 1/2$.
5. Step 5: Substitute the limits of integration into the result. Compute $(1/3)\; *\; [arcsin(1/2)\; -\; arcsin(0)]$. Recall that $arcsin(1/2)$ and $arcsin(0)$ are standard values from trigonometry.

Key Concepts

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫ from a to b f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values over the interval [a, b].

Substitution Method

The substitution method is a technique used to simplify the process of integration by changing the variable of integration. This involves substituting a new variable for a function of the original variable, which can make the integral easier to evaluate. It is particularly useful when dealing with integrals that involve composite functions or complicated expressions.

Trigonometric Substitution

Trigonometric substitution is a specific technique used in integration to simplify integrals involving square roots of quadratic expressions. By substituting a trigonometric function for a variable, such as x = (1/3)sin(θ) for √(1 - 9x²), the integral can often be transformed into a more manageable form. This method leverages the Pythagorean identities to facilitate the integration process.