Problem 37

Question

Use a CAS to find an antiderivative, then verify the answer by computing a derivative. (a) $\int x^{2} e^{-x^{3}} d x$ (b) $\int \frac{1}{x^{2}-x} d x$ (c) $\int \csc x \, d x$

Step-by-Step Solution

Verified

Answer

The antiderivatives of the functions $(a) x^{2} e^{-x^{3}}, (b) 1/(x^{2}-x)$, and $(c) \csc{x}$ are $-e^{-x^{3}}/3 + C, ln|x| - x + C$, and $- \ln|\csc{x} + \cot{x}| + C$ respectively.

1Step 1: Part (a) - Find the Antiderivative of $x^{2} e^{-x^{3}}$

First, let's find the antiderivative of $x^{2} e^{-x^{3}}$. We'll be using integration by substitution, let's set $u = -x^{3}$. The derivative of $u$ with respect to $x$ is $-3x^{2}$, which we will denote as $du = -3x^{2}dx$. Rearranging for $dx$, we get $dx = -du / (3x^{2})$. The integral then becomes $\int -e^{u} du / 3$, which simplifies to $-e^{u}/3 + C$, where $C$ is the constant of integration. Substituting back $u = -x^{3}$, we get the antiderivative as $-e^{-x^{3}}/3 + C$.

2Step 2: Part (a) - Verify the Antiderivative by Differentiation

To verify the antiderivative, we differentiate it and check if we obtain the original function. Differentiating the antiderivative $-e^{-x^{3}}/3 + C$ with respect to $x$ gives $(3x^{2}e^{-x^{3}})/3 = x^{2}e^{-x^{3}}$, which is the original function, thus confirming our antiderivative is correct.

3Step 3: Part (b) - Find the Antiderivative of $1/(x^{2}-x)$

In the second part, we need to find the antiderivative of $1/(x^{2}-x)$. This requires being familiar with the basic integral definitions. The function can be rewritten as $\int (1/x - 1) dx$. The integral of $1/x$ is $ln|x|$ and the integral of $-1$ is $-x$. Therefore, the antiderivative is $ln|x| - x + C$.

4Step 4: Part (b) - Verify the Antiderivative by Differentiation

We verify by differentiation. We take the derivative of the antiderivative $ln|x| - x + C$ which yields $1/x - 1$, the original function, confirming our antiderivative is correct.

5Step 5: Part (c) - Find the Antiderivative of $\csc{x}$

Lastly, for part c, we need to find the antiderivative of $\csc{x}$. For this integral, knowing the result of higher derivatives of trig functions such as $\csc{x}$ and $\cot{x}$ is required. The integral of $\csc{x}$ is $- \ln|\csc{x} + \cot{x}| + C$.

6Step 6: Part (c) - Verify the Antiderivative by Differentiation

The derivative of the antiderivative $- \ln|\csc{x} + \cot{x}| + C$ is $\csc{x}$, which is the original function, thus confirming our antiderivative is correct.

Key Concepts

Integration by SubstitutionIntegration of Rational FunctionsIntegration of Trigonometric Functions

Integration by Substitution

Integration by substitution is like reverse chain rule. We use it when the integral includes a composite function. The goal is to simplify the integral by changing the variable. Let's see how it works using the example of $ \int x^2 e^{-x^3} \, dx $.

First, identify the inner function. Here, it's $-x^3$. Set $u = -x^3$.
Differentiate $u$ to find $du$. We have $du = -3x^2 \, dx$.
Solve for $dx$ to get $dx = \frac{-du}{3x^2}$.
Substitute into the original integral, changing all $x$-terms to $u$-terms.This results in $\int -e^u \, \frac{du}{3}$.
Simplify and integrate: $-\frac{e^u}{3} + C$.
Finally, substitute back $u = -x^3$ to express the result in terms of $x$.

This process simplifies complex integrals into more basic forms that are easier to solve.

Integration of Rational Functions

Rational functions are ratios of polynomials. Integrating them involves strategies like partial fraction decomposition. Consider $ \int \frac{1}{x^2 - x} \, dx $.

The function $ \frac{1}{x^2 - x} $ can be split through partial fractions.
Rewrite as $ \int \left( \frac{1}{x} - 1 \right) \, dx $.
Integrate each term separately. The antiderivative of $ \frac{1}{x} $ is $ \ln|x| $.
The antiderivative of $-1$ is $-x$.
Combine: $ \ln|x| - x + C $.

By breaking the function into simpler parts, integration becomes straightforward, relying on familiar rules.

Integration of Trigonometric Functions

Integrating trigonometric functions often requires knowledge of specific integral formulas. The integral of $ \csc{x} $ is a classic example.

This integral, $ \int \csc{x} \, dx $, results in: $- \ln|\csc{x} + \cot{x}| + C$.
Knowing the derivatives of $\csc{x}$ and $\cot{x}$ helps in memorizing this result.
To verify, differentiate $- \ln|\csc{x} + \cot{x}| + C$ to check it equals $\csc{x}$.
This step ensures our integration process is correct.

Recognizing standard integrals for trigonometric functions assists in tackling these specific cases efficiently.

Problem 37

Problem 38

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