Problem 17
Question
Evaluate the integral. $$\int \frac{3 x^{3}}{x^{4}+5} d x$$
Step-by-Step Solution
Verified Answer
\(3/4 * ln|x^4 + 5| + c\)
1Step 1: Identify the substitution
Choose \(u\) as the denominator, \(x^4+5\). Now we differentiate \(u\) to get \(du\) and we get \(du = 4x^3dx\).
2Step 2: Change the integral in terms of \(u\)
We rearrange the \(du\) equation to solve for \(dx\), resulting in \(dx = du / (4x^3)\). Then substitute \(u\) and \(dx\) into the integral: \(\int 3x^3 / u \cdot du / (4x^3)\). This simplifies to \(3/4 * \int du/u\).
3Step 3: Evaluation of integral
The integral of \(1/u\) with respect to \(u\) is \(ln|u|\), so the integral becomes \(3/4 * ln|u| + c\), where \(c\) represents the constant of integration.
4Step 4: Substitute back for \(u\)
Returning to the original variable, we substitute \(u = x^4 + 5\) back into the integral, so the solution is \(3/4 * ln|x^4 + 5| + c\).
Key Concepts
U-SubstitutionIndefinite IntegralIntegration TechniquesNatural Logarithm
U-Substitution
When faced with an integral that is not straightforward to solve, u-substitution comes as a powerful technique to simplify the problem. This technique involves choosing a part of the integrand (the expression inside the integral) and substituting it with a new variable, typically u. The purpose is to convert the integral into a simpler form that is easier to evaluate.
As seen in the exercise solution, by selecting u = x^4 + 5, the integral transforms into a much simpler form. It’s important to express dx in terms of du to complete the substitution. U-substitution can often make the difference between an intractable problem and one with a clear path to the solution.
As seen in the exercise solution, by selecting u = x^4 + 5, the integral transforms into a much simpler form. It’s important to express dx in terms of du to complete the substitution. U-substitution can often make the difference between an intractable problem and one with a clear path to the solution.
Indefinite Integral
The indefinite integral is the antiderivative of a function, which represents a family of functions that differ by a constant. In the context of the exercise, the indefinite integral is written as \(\frac{3 x^{3}}{x^{4}+5} dx\).
Finding the Indefinite Integral
Approaching an indefinite integral typically involves identifying patterns or applying specific techniques, like u-substitution, to find the antiderivative. Always remember that the result will include a + c, representing an arbitrary constant, since any derivative of a constant is zero, making it impossible to determine from the derivative alone.Integration Techniques
There's a variety of integration techniques used in calculus to solve integrals, with each method suitable for different types of integrands. In addition to u-substitution, techniques include integration by parts, partial fraction decomposition, trigonometric substitution, and others.
Each method requires a certain familiarity to identify when and how to use it effectively. For instance, recognizing when to apply u-substitution involves looking for functions within your integrand that whose derivative is also present. In our original exercise, the derivative of u = x^4+5, which is 4x^3dx, appears simplifying the integrand significantly after substitution.
Each method requires a certain familiarity to identify when and how to use it effectively. For instance, recognizing when to apply u-substitution involves looking for functions within your integrand that whose derivative is also present. In our original exercise, the derivative of u = x^4+5, which is 4x^3dx, appears simplifying the integrand significantly after substitution.
Natural Logarithm
The natural logarithm, denoted as ln, is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.71828. In integration, the natural logarithm is particularly important because it is the antiderivative of 1/x.
In the exercise, once the integral is simplified via u-substitution, we encounter the simple integral \(\int\frac{du}{u}\), the antiderivative of which is ln|u| + c. It's important to include the absolute value around u to account for values of u that could be negative, as the logarithm is only defined for positive arguments. This understanding of the natural logarithm is vital for solving integrals that result in this form.
In the exercise, once the integral is simplified via u-substitution, we encounter the simple integral \(\int\frac{du}{u}\), the antiderivative of which is ln|u| + c. It's important to include the absolute value around u to account for values of u that could be negative, as the logarithm is only defined for positive arguments. This understanding of the natural logarithm is vital for solving integrals that result in this form.
Other exercises in this chapter
Problem 16
Use summation rules to compute the sum. $$\sum_{i=1}^{250}\left(i^{2}+8\right)$$
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Fill in the blanks with the most appropriate power of 2(2,4,8 etc.). If you double \(n,\) the error in the Midpoint Rule is divided by____ \(\quad\) If you doub
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Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{4} x(x-2) d x$$
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