Question

Evaluate the integrals in Exercises 67–74 in terms ofb. natural logarithms.71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))

Accepted Answer

Recognize that the integral has the form $$ \int \frac{dx}{x \sqrt{1 - a^2 x^2}} $$ where $$ a = 4 $$ because $$ 16x^2 = (4x)^2 . $$This suggests a substitution related to inverse hyperbolic or trigonometric functions, but since the problem asks for an expression in terms of natural logarithms, we will use an appropriate substitution to rewrite the integral accordingly. Use the substitution $$ t = \sqrt{1 - 16x^2} . $$Then, differentiate both sides to express $$ dx  in $$terms of $$ dt $$ and $$ x . $$This will help rewrite the integral in terms of $$ t $$ and simplify the square root expression. Rewrite the integral limits from $$ x = \frac{1}{5} $$ and $$ x = \frac{3}{13}  to $$the corresponding values of $$ t $$ using the substitution $$ t = \sqrt{1 - 16x^2} . $$This is necessary to evaluate the definite integral after substitution. After substitution, simplify the integral to a form involving $$ \frac{dt}{1 - t^2}  or a $$similar rational function that can be integrated into a natural logarithm expression using partial fractions or standard integral formulas. Integrate the simplified expression to obtain a result in terms of natural logarithms, then substitute back to the original variable $$ x $$ and apply the limits to express the definite integral fully in terms of natural logarithms.

Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))

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Key Concepts

Integration involving square roots of quadratic expressions

Trigonometric substitution

Expressing results in terms of natural logarithms