Question

Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrals before applying the suggested technique of integration. You do not need to evaluate the integrals. $$\int (1+\tan x){\sec }^{2}xdx$$

Short answer

The integral $\int (1+\tan x){\sec }^{2}xdx$ is simplified into two separate integrals: $\int {\sec }^{2}xdx$ and $\int \tan x{\sec }^{2}xdx$. The integration technique used for $\int {\sec }^{2}xdx$ is directly applying the antiderivative rule, while the integration technique used for $\int \tan x{\sec }^{2}xdx$ is substitution method with $u=\tan x$.

Step-by-step solution

Distribute the secant squared function

With the given integral, we can distribute the ${\sec }^{2}x$ term to both parts inside the parenthesis: $$\int (1+\tan x){\sec }^{2}xdx=\int ({\sec }^{2}x+\tan x{\sec }^{2}x)dx$$ Now that we have simplified the integral, let's identify an integration technique for each part of the sum:

Identify integration technique for the first term

For the first term, $\int {\sec }^{2}xdx$, we can directly apply the definition of the antiderivative as it is a well-known rule that: $$\int {\sec }^{2}xdx=\tan x+C$$

Identify integration technique for the second term

For the second term, $\int \tan x{\sec }^{2}xdx$, we apply the substitution method. Let $u=\tan x$. Then, the derivative of $u$ with respect to $x$ is: $$\frac{du}{dx}={\sec }^{2}x\Rightarrow du={\sec }^{2}xdx$$ By making the substitution and changing the integral, we have: $$\int \tan x{\sec }^{2}xdx=\int udu$$ We will find an antiderivative of `u` with respect to `x`. In summary: - For $\int {\sec }^{2}xdx$, the integration technique used is directly applying the antiderivative rule. - For $\int \tan x{\sec }^{2}xdx$, the integration technique used is substitution method with $u=\tan x$.

Key concepts

Integration by Substitution

The integration by substitution method, also known as the u-substitution, is a powerful technique in calculus for finding the antiderivatives of more complex functions. It involves replacing a part of the integrand with a new variable, typically denoted as 'u'.

For instance, if we have a composite function like a trigonometric expression multiplied by its own derivative, we look for a substitution that will simplify the integral. Once you've determined a suitable substitution, you find the differential of 'u' (denoted as du) and replace the corresponding parts of the integrand with 'u' and 'du'.

This technique not only simplifies the integral but also reveals familiar patterns that are easier to integrate. After integrating with respect to 'u', you then substitute back the original variables to complete the problem.

Antiderivatives

Antiderivatives are the reverse process of differentiation. Simply put, if differentiation gives you the speed of a car at any moment, antiderivatives tell you the position of the car over time, provided you know where you started (this is the '+ C' in the integrals).

When you take the antiderivative of a function, you're essentially looking for a function whose derivative is the original function you started with. In the context of the integral of ${\sec }^{2}xdx$, we know from calculus that the antiderivative of ${\sec }^{2}x$ is $\tan x$ because the derivative of $\tan x$ gives us ${\sec }^{2}x$. Therefore, $\int {\sec }^{2}xdx=\tan x+C$, where 'C' represents the constant of integration.

Secant Squared Function

In trigonometry, the secant function, denoted $\sec x$, is defined as the reciprocal of the cosine function. The secant squared function is the square of the secant function, ${\sec }^{2}x$, and it plays a significant role in integration. An integral involving ${\sec }^{2}x$ often arises in problems involving trigonometric identities.

One of the key aspects to remember is that the derivative of the tangent function, $\tan x$, is ${\sec }^{2}x$. This relationship makes finding antiderivatives of ${\sec }^{2}x$ straightforward, since we readily know it will lead to $\tan x+C$. Recognizing these kinds of relationships between trigonometric functions is an essential skill when solving integrals that involve trigonometric terms.

Trigonometric Integration

Trigonometric integration involves finding the antiderivatives of trigonometric functions. This area of integration can be tricky due to the cyclical nature of trigonometric functions, but it is manageable with knowledge of some key insights and identities. When faced with an integral that includes trigonometric functions, look for opportunities to apply these identities to simplify the problem.

Trigonometric integrals often involve strategies such as substituting trigonometric identities, using u-substitution, or rewriting the integrands in a more workable form. For example, recognizing that $\int {\sec }^{2}xdx$ simplifies to $\tan x+C$ or that $\int \sin x\cos xdx$ can be simplified by substitution are key steps in solving trigonometric integrals. Mastery of these techniques opens the door to solving a wide variety of integrals involving trigonometric functions.