Summary
Highlights
The first problem involves evaluating the integral of dx / (4x² + 9). This integral follows the form u² + a², where u = 2x and a = 3. The appropriate trigonometric substitution for this case is u = a tan θ, so 2x = 3 tan θ. From this, expressions for dx and the denominator (4x² + 9) are derived in terms of θ. After substitution and simplification, the integral becomes ∫ (1/6) dθ, which simplifies to (1/6)θ + C. Finally, θ is substituted back with arctan(2x/3) to give the final answer: (1/6)arctan(2x/3) + C.
The second problem involves an integral of the form ∫ x² dx / √(4 - x²). This fits the a² - u² case, where a = 2 and u = x. The substitution is u = a sin θ, so x = 2 sin θ. Expressions for x², dx, and √(4 - x²) are developed in terms of θ using a right triangle. After substitution, trigonometric identities (specifically the half-angle identity for sin²θ) are used to simplify the integral. The integral is solved for θ, and then the result is converted back to x using inverse trigonometric functions and the relationships from the right triangle. The final answer is 2arcsin(x/2) - x√(4 - x²)/2 + C.
The third problem involves evaluating ∫ x³√(4x² - 1) dx, which matches the u² - a² form. Here, u = 2x and a = 1. The appropriate trigonometric substitution is u = a sec θ, so 2x = sec θ. Derivations for x, dx, and √(4x² - 1) are made based on this substitution and a right triangle. The integral is then transformed into an expression involving secants and tangents. After several steps of algebraic and trigonometric simplification, including power reduction and u-substitution (u = tan θ), the integral is solved. The result, expressed in terms of u, is then converted back to x to find the final answer: (4x² - 1)^(5/2) / 80 + (4x² - 1)^(3/2) / 48 + C.
The final problem addresses ∫ x² dx / (x² + 16)^(3/2), which falls under the u² + a² case. The substitution here is u = a tan θ, meaning x = 4 tan θ. Through a right triangle and trigonometric relationships, expressions for x, dx, and (x² + 16) are established in terms of θ. The integral is transformed and simplified using trigonometric identities, eventually resolving to ∫ (sec θ - cos θ) dθ. The integration yields ln|sec θ + tan θ| - sin θ + C. Finally, this expression is converted back to x using the relationships derived from the right triangle: ln|√(x² + 16)/4 + x/4| - x/√(x² + 16) + C. This can be further simplified using logarithm properties.