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Magazine Chapter 5: Problem 26
Evaluate the integrals using appropriate substitutions. $$ \int \frac{x}{\sqrt{4-5 x^{2}}} d x $$
Short Answer
Expert verified
The integral evaluates to \( -\frac{1}{5} \sqrt{4 - 5x^2} + C \).
Step by step solution
01
Identify Substitution
We will use the substitution method to solve the integral. Notice that the integral has a term under a square root, suggesting a trigonometric or hyperbolic substitution may be appropriate. Identify the term under the square root:\( u = 4 - 5x^2 \). This gives \( du = -10x \, dx \).
02
Solve for dx
Rewrite the derivative \( du = -10x \, dx \) to solve for \( dx \):\[ dx = \frac{du}{-10x} \].
03
Substitute Expressions
Substitute \( u = 4 - 5x^2 \) and \( dx = \frac{du}{-10x} \) into the original integral:\[ \int \frac{x}{\sqrt{u}} \cdot \frac{du}{-10x} \].
04
Simplify the Integral
Cancel out the \( x \) terms from the integral, and move constants outside the integral:\[ \int \frac{1}{\sqrt{u}} \cdot \left(\frac{du}{-10}\right) = -\frac{1}{10} \int \frac{1}{\sqrt{u}} \, du \].
05
Integrate
Integrate using the formula \( \int u^{-1/2} \, du = 2u^{1/2} + C \):\[ -\frac{1}{10} \cdot 2u^{1/2} = -\frac{1}{5}u^{1/2} + C \].
06
Return to Original Variable
Substitute back \( u = 4 - 5x^2 \) to get the expression in terms of \( x \):\[ -\frac{1}{5} \sqrt{4 - 5x^2} + C \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a powerful tool in integral calculus that simplifies the process of integrating complex functions. It involves changing variables to make an integral easier to evaluate.
A common case is to substitute a variable with a function that is inside another function, often under a square root or within a polynomial.
Here, the basic idea is to replace a difficult function with a simpler one, followed by adjusting the differentials correspondingly. Some steps usually involved in substitution are:
Here, the basic idea is to replace a difficult function with a simpler one, followed by adjusting the differentials correspondingly. Some steps usually involved in substitution are:
- Identify a portion of the integral that can be substituted to simplify the function. This is often a candidate for derivative chains inside the integral.
- Choose an appropriate substitution, such as setting a variable equal to an expression found in the integral.
- Differentiate the substitution with respect to the original variable.
- Replace all instances of the chosen variable and differential in the integral.
- Once integration is complete, substitute back the original variables.
Definite Integrals
Definite integrals can be thought of as the accumulation of quantities, such as areas under a curve. Unlike indefinite integrals, which represent a family of functions, definite integrals provide a precise numerical value.
The notation for a definite integral is typically written with limits of integration on the integral sign, for instance: \[\int_{a}^{b} f(x) \, dx\]This would calculate the net area between the curve of \( f(x) \) from \( x = a \) to \( x = b \). The process requires:
The notation for a definite integral is typically written with limits of integration on the integral sign, for instance: \[\int_{a}^{b} f(x) \, dx\]This would calculate the net area between the curve of \( f(x) \) from \( x = a \) to \( x = b \). The process requires:
- Determining the antiderivative or integral of the function.
- Evaluating this antiderivative at the upper and lower limits of integration.
- Subtracting the evaluated results (\( F(b) - F(a) \)) to find the accumulated value.
Trigonometric Substitution
Trigonometric substitution is a specialized technique for integrating functions involving square roots. This method can transform a difficult integral into a simpler trigonometric form.
This is incredibly useful when dealing with integrands that include expressions such as \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), or \( \sqrt{x^2 - a^2} \).The procedure typically involves:
This is incredibly useful when dealing with integrands that include expressions such as \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), or \( \sqrt{x^2 - a^2} \).The procedure typically involves:
- Identifying the form of the expression under the square root and choosing the appropriate trigonometric substitution. Common substitutions include \( x = a \sin(\theta) \), \( x = a \tan(\theta) \), or \( x = a \sec(\theta) \).
- Replacing \( x \) and \( dx \) with their trigonometric counterparts.
- Simplifying the integral using trigonometric identities.
- Solving the integral in terms of the trigonometric function.
- Converting back to the original variable once integration is complete.
