Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \tan (4 x+2) d x$$

Short answer

\(-\frac{1}{4}\ln|\cos(4x+2)|+ C\)

Step-by-step solution

Setting the Substitution

Set \(u = 4x + 2\). So, the differential \(du\) is \(du = 4 dx\). To match the integral, divide both sides by 4, giving \(dx = \frac{1}{4} du\).

Substituting into the Integral

Replace \(4x + 2\) with \(u\) and \(dx\) with \(\frac{1}{4} du\). This gives: \(\frac{1}{4}\int \tan(u) du\). The 1/4 is a constant coefficient, so it's factored out of the integral.

Solve the Integral

The integral of the tangent function is well known and is given by \(-\ln |\cos(u)|\). So, \(\frac{1}{4} \int \tan(u) du \) becomes \(-\frac{1}{4}\ln |\cos(u)|+ C\), where C represents the constant of integration.

Replace \(u\) with Original Variable

Finally, replace \(u\) with \(4x+2\) to convert the solution back into the terms of the original variable \(x\), giving: \(-\frac{1}{4}\ln |\cos(4x+2)|+ C\).

Key concepts

Indefinite Integral

An indefinite integral, also known as an antiderivative, represents the family of functions that gives the original function when differentiated. If you're given the function \( f(x) = \tan(4x + 2) \), then the indefinite integral is the collection of all functions \( F(x) \) such that \( F'(x) = \tan(4x + 2) \).

This is generally represented as \( \int f(x) \, dx = F(x) + C \), where \( C \) is the 'constant of integration'. It denotes that there could be an infinite number of functions that differ by a constant value all of which are solutions to the integral. When solving for an indefinite integral, it is like unwinding the process of differentiation and figuring out the original function from its rate of change.

U-Substitution Method

The u-substitution method, also known as the change of variables, is a technique used to simplify the process of finding the antiderivative of more complex functions. When faced with an integral that is not in standard form, substitution allows for the re-expression of the integral in terms of a new variable \( u \), which often transforms the problem into one with a known solution.

The key to this method is choosing \( u \) in such a way that \( du \) (the differential of \( u \) with respect to \( x \) ) corresponds to a part of the integral we are trying to solve. As seen in the exercise, after setting \( u = 4x + 2 \), the term \( dx \) is expressed in terms of \( du \) by dividing by the derivative of \( u \) with respect to \( x \), creating a simpler integral in terms of \( u \). This process streamlines finding the integral by working with a reduced and often more familiar form.

Antiderivatives of Trigonometric Functions

The antiderivatives, or integrals, of trigonometric functions are a significant part of calculus. Due to the periodic nature of trigonometric functions, their antiderivatives are frequently encountered in various mathematical problems.

For instance, the antiderivative of \( \tan(x) \) is \( -\ln |\cos(x)| \) up to an arbitrary constant. This fundamental result is crucial for solving integrals involving the tangent function, like the one seen in the exercise. Trigonometric antiderivatives often involve logarithmic functions due to the inverse relationships present in their derivatives. Knowing these antiderivatives is key for easing the integration process of trigonometric expressions.

Constant of Integration

The constant of integration, represented by \( C \) in integrals, is essential in the solution of indefinite integrals. Because differentiation erases any constant value (since the derivative of any constant is zero), when calculating an indefinite integral, we must include an arbitrary constant to account for this 'lost' information.

In practice, the constant of integration symbolizes the infinite set of antiderivative functions that are possible. After solving an indefinite integral, it's mandatory to add \( C \) to cover all potential original functions. In the solution provided for \( \int \tan(4x + 2) \, dx \), the final expression must include \( C \) to be completely correct. This constant becomes determined when an initial condition or boundary value is known, moving from indefinite to definite calculus.