Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. More than one integration method can be used to evaluate $\int \frac{dx}{1-x^2}$ b. Using the substitution $u=\sqrt[3]{x}$ in $\int \sin \sqrt[3]{x}dx$ leads to $\int 3u^2\sin udu$ c. The most efficient way to evaluate $\int \tan 3x{\sec }^{2}3xdx$ is to first rewrite the integrand in terms of $\sin 3x$ and $\cos 3x$ d. Using the substitution $u=\tan x$ in $\int \frac{{\tan }^{2}x}{\tan x-1}dx$ leads to $\int \frac{u^2}{u-1}du$

Short answer

a. More than one integration method can be used to evaluate $\int \frac{dx}{1-x^2}$. b. Using substitution $u=\sqrt[3]{x}$ in $\int \sin \sqrt[3]{x}dx$ leads to $\int 3u^2\sin udu$. c. The most efficient way to evaluate $\int \tan 3x{\sec }^{2}3xdx$ is to first rewrite the integrand in terms of $\sin 3x$ and $\cos 3x$. d. Using substitution $u=\tan x$ in $\int \frac{{\tan }^{2}x}{\tan x-1}dx$ leads to $\int \frac{u^2}{u-1}du$. A: Statements a and b are true, while statements c and d are false.

Step-by-step solution

Identify possible integration methods for this integral

Two integration methods that can be applied here are substitution and partial fraction decomposition. 1. Substitution: We can use the substitution $x=\sin u$, which will give us $dx=\cos udu$. Replacing $x$ and $dx$ in the integral, we get: $\int \frac{\cos udu}{1-{\sin }^{2}u}=\int \frac{\cos udu}{{\cos }^{2}u}$ 2. Partial Fraction Decomposition: We can rewrite the integrand as $\frac{1}{(1+x)(1-x)}$. This method is most effective when the denominator can be factored into linear factors. a. Result: The given statement is true. More than one integration method can be used to evaluate $\int \frac{dx}{1-x^2}$. b. Using substitution $u=\sqrt[3]{x}$ in $\int \sin \sqrt[3]{x}dx$ leads to $\int 3u^2\sin udu$

Perform the given substitution

If $u=\sqrt[3]{x}$, then $x=u^3$ and $dx=3u^{2}du$. Now, replace $x$ and $dx$ with these expressions in the given integral: $\int \sin \sqrt[3]{x}dx=\int \sin u\cdot 3u^{2}du=\int 3u^2\sin udu$ b. Result: The given statement is true. The substitution leads to the integral $\int 3u^2\sin udu$. c. The most efficient way to evaluate $\int \tan 3x{\sec }^{2}3xdx$ is to first rewrite the integrand in terms of $\sin 3x$ and $\cos 3x$

Check if rewriting the integrand in terms of $\sin 3x$ and $\cos 3x$ is efficient

The given integral can be written as $\int \frac{\sin 3x}{\cos 3x}\cdot \frac{1}{{\cos }^{2}3x}dx$. Multiplying the two fractions results in $\int \frac{\sin 3x}{{\cos }^{3}3x}dx$. However, this transformation does not provide any obvious simplification or easier integration method. An alternative approach would be to use substitution. Letting $u=\tan 3x$, we have $du=3{\sec }^{2}3xdx$. The integral becomes $\int \frac{1}{3}udu$, which is much simpler to evaluate. c. Result: The given statement is false. Rewriting the integrand in terms of $\sin 3x$ and $\cos 3x$ does not provide an efficient way to evaluate the integral. Using the substitution method is more efficient. d. Using substitution $u=\tan x$ in $\int \frac{{\tan }^{2}x}{\tan x-1}dx$ leads to $\int \frac{u^2}{u-1}du$

Perform the given substitution

If $u=\tan x$, then $du={\sec }^{2}xdx$. The integral becomes: $\int \frac{u^2}{u-1}{\sec }^{2}xdx=\int \frac{u^2}{u-{\sec }^{2}x}du$ The substitution does not lead to the integral $\int \frac{u^2}{u-1}du$. d. Result: The given statement is false. Using the substitution $u=\tan x$ in $\int \frac{{\tan }^{2}x}{\tan x-1}dx$ does not lead to $\int \frac{u^2}{u-1}du$.

Key concepts

Substitution Method

The substitution method is a powerful tool in calculus, particularly useful for simplifying an integral by changing the variable of integration. This method requires you to choose a new variable that will transform the integrand into a form that is easier to integrate.

Let's examine a classic example: Suppose we have an integral of the form $\int \sin \sqrt[3]{x}dx$. By letting $u=\sqrt[3]{x}$, we can rewrite the integral in terms of $u$, which simplifies the integrand and makes the integral more manageable. In this case, we also need to change $dx$ to $du$ accordingly, which involves computing the derivative of $u$ with respect to $x$ to obtain $dx=3u^{2}du$.

When using the substitution method, always check the entire integrand to ensure all instances of the original variable and its differentials are replaced with the new ones. If done correctly, you should end up with an integral solely in terms of your new variable $u$, like $\int 3u^2\sin udu$, which is often much easier to evaluate.

Partial Fraction Decomposition

Partial fraction decomposition is an algebraic technique used to break down complex rational expressions into simpler ones that are easier to integrate. This method is particularly useful when dealing with integrals where the denominator can be factored into linear or quadratic factors that do not repeat.

For instance, consider the integral $\int \frac{dx}{1-x^2}$. The denominator $1-x^2$ can be factored into $1+x$ and $1-x$. Using partial fraction decomposition, we can express the original fraction as a sum of simpler fractions:$$\frac{A}{1+x}+\frac{B}{1-x}$$ where $A$ and $B$ are constants that can be determined through various techniques like equating coefficients or plugging in convenient values for $x$.

Once the decomposition is complete, the integral becomes the sum of integrals of these simpler fractions, which are typically straightforward to evaluate using the basic set of antiderivatives known in calculus.

Trigonometric Integration

Trigonometric integration involves the integration of trigonometric functions and often requires specific techniques or identities to simplify the integrals. One common approach is to rewrite the integrand in terms of basic trigonometric identities, such as ${\sin }^2(x)+{\cos }^2(x)=1$, or to use substitution to convert the integral into a more direct trigonometric form.

For the integral $\int \tan 3x{\sec }^{2}3xdx$, directly applying a trigonometric identity might not lead to a simpler form. A more efficient method might be to use a substitution like $u=\tan (3x)$, which simplifies the process. This yields $du=3{\sec }^2(3x)dx$, and the integral becomes $\int \frac{1}{3}udu$, greatly simplifying the problem.

It is vital to recognize when to employ trigonometric identities and when a substitution will lead to a simpler integral. Mastery of trigonometric integrals typically requires familiarity with a range of trigonometric formulas and practice with strategic substitution choices.