Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \sec \left(\theta+\frac{\pi}{2}\right) \tan \left(\theta+\frac{\pi}{2}\right) d \theta$$

Short answer

\(\sec(\theta + \frac{\pi}{2}) + C\)

Step-by-step solution

Identify the Substitution

From the integral, \(\int \sec \left(\theta+\frac{\pi}{2}\right) \tan \left(\theta+\frac{\pi}{2}\right) d \theta\), choose the substitution \(u = \theta + \frac{\pi}{2}\). This will simplify the argument of the sec and tan functions.

Compute the Differential

To perform a substitution, we need to know how the differential \(du\) relates to the original variable \(d\theta\). Since \(u = \theta + \frac{\pi}{2}\), we have that \(du = d\theta\).

Substitute Back Into the Integral

Replace \(\theta\) with \(u - \frac{\pi}{2}\) and \(d\theta\) with \(du\) in the original integral. This gives us the new integral \(\int \sec u \tan u du\).

Evaluate the Integral

The integral \(\int \sec u \tan u du\) is a standard integral, whose result is \(\sec u + C\)). Substituting \(u = \theta + \frac{\pi}{2}\) back in, the final solution to the integral is \(\sec(\theta + \frac{\pi}{2}) + C\).

Key concepts

Integral Calculus

Integral calculus is a fundamental branch of mathematics focused on finding the accumulative totals, such as areas under curves and the total accumulation of a quantity over a space. Integrals can be indefinite or definite—where an indefinite integral yields a family of functions (the antiderivatives), a definite integral computes a specific numerical value bounded by limits.

It plays a crucial role not only in pure mathematics but also in physics, engineering, economics, and other fields. Many functions, particularly those that describe physical phenomena, are described in terms of their rates of change—their derivatives—and integral calculus allows us to work backwards from the derivative to the original function or to 'sum up' an infinite number of infinitesimal contributions.

Trigonometric Integrals

Trigonometric integrals involve the integration of trigonometric functions such as sine, cosine, tangent, and their reciprocals. These functions frequently appear in problems involving periodic phenomena or right triangles.

To solve trigonometric integrals, certain strategies can be employed, such as using trigonometric identities to simplify the expression before integrating, or sometimes breaking them into simpler parts using partial fractions. These strategies allow us to rewrite complex trigonometric expressions into forms that can be integrated more directly.

U-Substitution

U-substitution is a technique used to simplify the process of finding an integral. It's much like the 'change of variables' method used in algebra but applied to calculus.

The idea is to make a substitution that simplifies the integral into a form that is easier to work with. Typically, you'll choose a part of the integrand (the expression inside the integral) to be a new variable, 'u', which simplifies the integral when you find the differential 'du'.

Guide to U-Substitution:

• Identify a function inside the integrand to be 'u'.
• Differentiate 'u' to find 'du'.
• Substitute 'u' and 'du' into the integral.
• Integrate with respect to 'u'.
• Replace 'u' with the original variable to find the answer.

Indefinite Integral

An indefinite integral, also known as an antiderivative, represents a family of functions whose derivative is the integrand. It is expressed without bounds, unlike the definite integral, and includes a constant of integration 'C' to represent all possible antiderivatives.

Indefinite integrals are written in the form \(\int f(x)\, dx = F(x) + C\), where \(F(x)\) is the antiderivative of \(f(x)\) and 'C' accounts for all the constant shifts along the y-axis that an antiderivative can have. In practice, finding the indefinite integral involves reversing the process of differentiation and applying integration techniques, such as u-substitution, to evaluate the integral. The resulting function describes all possible shapes of antiderivatives that have the same rate of change, represented by the original function 'f(x)'.