91Ó°ÊÓ

When we talk about an indefinite integral, we're essentially discussing finding the antiderivative of a function. Unlike definite integrals, indefinite integrals do not have specified limits of integration, and as a result, they represent a family of functions rather than a single number. The solution is often expressed with an arbitrary constant of integration, typically denoted as C.

This constant is crucial because when differentiating the antiderivative, it vanishes. Therefore, each indefinite integral includes this constant to account for every possible solution. For instance, in our exercise, the indefinite integral of the original function \[\int \frac{1}{(25+4x^2)^{3/2}} \ dx\]was evaluated using trigonometric substitution. After substitution and integration of \(\cos(\theta)\), the result is given and returned to terms of x with that constant included:\[\frac{2x}{25\sqrt{4x^2+25}} + C.\]This tells us about the entire set of antiderivatives for the original function.

The Pythagorean identity is a fundamental trigonometric concept that drives many transformations, particularly in substitutions used during calculus problems like this one. It stems from the Pythagorean theorem and informs us that the sum of squares of the sine and cosine of an angle equals one:

• \(\sin^2(\theta) + \cos^2(\theta) = 1\)
• \(\sec^2(\theta) = 1 + \tan^2(\theta)\)

This second form is particularly relevant when dealing with expressions that fit the format seen in the integral problem.

In our exercise, this identity allows us to transform the expression \(25 + 4x^2\) into \(25 \sec^2(\theta)\) when using the substitution \(x = \frac{5}{2} \tan(\theta)\). This simplification and use of the identity helps in converting the problem into a more easily solvable form in terms of trigonometric functions.
By understanding how the Pythagorean identity works in concert with substitutions, you can tackle a wide range of integrals involving trigonometric transformations.

In mathematics, particularly trigonometry and calculus, understanding right triangles is crucial, especially for substitution problems. When we encounter a problem that involves trigonometric functions like tangent or sine, visualizing or sketching a right triangle can simplify understanding the relationship between those functions and the given variables.

In the exercise, where we have \(\tan(\theta) = \frac{2x}{5}\), drawing a right triangle aids in finding \(\sin(\theta)\). Here:

• The opposite side to \(\theta\) is \(2x\).
• The adjacent side is \(5\).

The hypotenuse, calculated using the Pythagorean theorem, becomes \(\sqrt{4x^2 + 25}\), allowing us to express\(\sin(\theta)\) as the ratio of the opposite to the hypotenuse:\[\sin(\theta) = \frac{2x}{\sqrt{4x^2 + 25}}.\]
Connecting right triangles with trigonometric identities and substitutions helps smoothly transition between different mathematical expressions and problem-solving approaches.