Question

In the following exercises, find each indefinite integral by using appropriate substitutions. $$ \int e^{\tan x} \sec ^{2} x d x $$

Short answer

\( e^{\tan x} + C \)

Step-by-step solution

Identify the Substitution

Look at the integral \( \int e^{\tan x} \sec^2 x \, dx \). The function \( e^{\tan x} \) suggests that \( \tan x \) is a good candidate for substitution. Let \( u = \tan x \).

Determine the Derivative

Find the derivative of \( u \) with respect to \( x \). Since \( u = \tan x \), \( \frac{du}{dx} = \sec^2 x \). This means \( du = \sec^2 x \, dx \).

Substitute into the Integral

Replace \( \tan x \) with \( u \) and \( \sec^2 x \, dx \) with \( du \) in the integral: \( \int e^{\tan x} \sec^2 x \, dx = \int e^u \, du \).

Integrate with Respect to \( u \)

The integral \( \int e^u \, du \) is a standard form. The integral of \( e^u \) with respect to \( u \) is \( e^u + C \), where \( C \) is the constant of integration.

Substitute Back to Original Variable

Substitute \( u = \tan x \) back into the result: \( e^u + C = e^{\tan x} + C \).

Write the Final Answer

The indefinite integral \( \int e^{\tan x} \sec^2 x \, dx \) is \( e^{\tan x} + C \).

Key concepts

Substitution Method

The substitution method in integration is a lot like unraveling a puzzle. When you encounter a complex integral, the trick is to find a substitution that simplifies it. In our example, we had the integral \( \int e^{\tan x} \sec^2 x \, dx \). The exponent \( e^{\tan x} \) suggests \( \tan x \) as a fitting candidate for substitution. You basically introduce a new variable \( u \) to replace \( \tan x \). This turns the integral, which looks a bit complicated, into something simpler, making it easier to solve.

• First, let \( u = \tan x \). This is the heart of the substitution technique.
• Next, find the derivative: \( \frac{du}{dx} = \sec^2 x \), which implies that \( du = \sec^2 x \, dx \).

Now, substitute \( \tan x \) with \( u \) and \( \sec^2 x \, dx \) with \( du \). The integral becomes \( \int e^u \, du \) — a more straightforward form that we can easily integrate.
Substitution can significantly simplify integration, especially when dealing with functions within functions, allowing you to perform integration more effectively.

Trigonometric Integrals

Trigonometric integrals can often seem intimidating at first, but with the right techniques, they become more manageable. In the given example, we dealt with \( \sec^2 x \), which is directly related to the trigonometric function \( \tan x \). This connection is what we use to our advantage.
Whenever you have trigonometric functions like \( \sin x, \cos x, \tan x \), and their derivatives, it's crucial to recognize how they relate to each other:

• For \( \sec^2 x \), we know that it is the derivative of \( \tan x \).
• This makes \( \sec^2 x \cdot dx \) a perfect candidate for the differential \( du \) when we set \( u = \tan x \).

Trigonometric integrals often require substitutions involving trigonometric identities or recognizing derivatives of trigonometric functions. With practice, spotting these relationships becomes second nature, aiding you in solving problems efficiently.

Integration Techniques

Integration techniques are essential tools in the calculus toolbox. For each type of problem, there is often a distinct method or strategy that simplifies finding the solution. In this example, combining substitution with knowledge of standard integral forms efficiently leads us to the solution.
One essential integration technique is recognizing and using standard integral forms:

• In the original problem, after substitution, \( \int e^u \, du \) is a standard form. The integral of \( e^u \) is simply \( e^u + C \), where \( C \) is the constant of integration.

After solving the integral, don't forget about substituting back to the original variable. This ensures the solution is in terms of the variable used in the initial problem. Practical application of these techniques allows us to address various integrals quickly and accurately, without the process feeling overwhelming.