Problem 20 Determine the integrals by makin... [FREE SOLUTION]

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Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

$Math Studyset 91影视 Explanations$ Math

14 Edition

Chapter 9: Problem 20

Determine the integrals by making appropriate substitutions. $$\int \frac{x^{2}}{3-x^{3}} d x$$

Short Answer

Expert verified

The integral is $-\frac{1}{3} \ln|3 - x^3| + C$.

Step by step solution

Identify a suitable substitution

Notice that the integrand has a function and its derivative. Let鈥檚 choose a substitution that simplifies the denominator. Choose $u = 3 - x^3$.

Compute the differential

Differentiate the substitution $u = 3 - x^3$ with respect to $x$: $du = -3x^2 dx$. Rearrange to express $dx$: $-\frac{1}{3} du = x^2 dx$.

Substitute in the integral

Substitute $u$ and $du$ into the integral: $\int \frac{x^2}{3-x^3} dx = \int \frac{-\frac{1}{3} du}{u}$.

Integrate with respect to $u$

Now integrate with respect to $u$: $-\frac{1}{3} \int \frac{1}{u} du$. The integral of $\frac{1}{u}$ is $\ln|u|$, so the result is $-\frac{1}{3} \ln|u| + C$.

Substitute back $u$

Replace $u$ with the original expression $3 - x^3$ to get the final answer: $-\frac{1}{3} \ln|3 - x^3| + C$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

integration by substitution

Integration by substitution is a method to simplify an integral by introducing a new variable. In this example, we use the substitution $u = 3 - x^3$. This helps transform a complicated integral into a simpler form. Here, our new variable $u$ replaces the expression $3 - x^3$. The goal is to find a suitable substitution that makes the integrand easier to handle.
To use substitution, follow these steps:

Choose a substitution that simplifies the integrand. Identify parts of the integrand that could be replaced.
Calculate the differential of your substitution. For this exercise, we calculate $du = -3x^2 dx$.
Transform the integral using your new variable and its differential. For example, substitute $u$ and $du$ into the integral.
Integrate with respect to the new variable. The integral becomes simpler, letting us use basic integration techniques.
Substitute the original expression back into the result. Convert back to the original variable to finalize the solution.

By mastering this technique, you can handle a wider range of integrals with confidence.

differentiation

Differentiation is the process of finding the derivative of a function. In this problem, we differentiated the substitution $u = 3 - x^3$. This gave us $du = -3x^2 dx$.
The derivative tells us how a function changes as its input changes. Here are key points about differentiation:

The derivative of $x^n$ is $nx^{n-1}$.
The derivative of a constant is zero.
The derivative of a sum is the sum of the derivatives.

In our example, we used the power rule to differentiate $3 - x^3$, resulting in $ 0 - 3x^2$. This step is crucial because it allows us to rewrite the integral in terms of $u$ and $du$. Knowing how to differentiate effectively can make the substitution step in integration much smoother.

natural logarithm

The natural logarithm, denoted as $\ln$, is the inverse function of the exponential function $e^x$. In our solution, we ended up with an integral involving $\frac{1}{u}$, which integrates to $\ln|u|$.
The natural logarithm has some important properties:

$\ln(ab) = \ln(a) + \ln(b)$
$\ln(a^b) = b \ln(a)$
The derivative of $\ln(x)$ is $\frac{1}{x}$

In this integral, we use the fact that the integral of $\frac{1}{u}$ is $\ln|u|$. After substituting back $u = 3 - x^3$, the solution becomes $-\frac{1}{3} \ln|3 - x^3| + C$. Understanding natural logarithms helps in integrating functions that turn into a logarithm after integration.

definite integrals

Definite integrals calculate the area under a curve over a specific interval. For definite integrals, we use limits of integration. In this exercise, the integral is indefinite (no limits).
To compute a definite integral:

Find the indefinite integral first.
Apply the limits of integration to the antiderivative.
Subtract the value of the antiderivative at the lower limit from the value at the upper limit.

Definite integrals are represented as $\int_{a}^{b} f(x) dx$. For example, if we had limits, say from $a$ to $b$, we would evaluate $F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$. While our current integral did not need definite integration, understanding how to handle limits is crucial for calculating exact areas and other practical applications.

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91影视

Determine the integrals by making appropriate substitutions. $$\int \frac{x^{2}}{3-x^{3}} d x$$

Short Answer

Step by step solution

Identify a suitable substitution

Compute the differential

Substitute in the integral

Integrate with respect to \(u\)

Substitute back \(u\)

Key Concepts

integration by substitution

differentiation

natural logarithm

definite integrals

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