Finding Indefinite Integrals
Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec² s/10 ds

Verified step by step guidance

1. Identify the integral to solve: \(\int \sec^{2}\left(\frac{s}{10}\right) \, ds\).
2. Recall the basic integral formula: \(\int \sec^{2}(x) \, dx = \tan(x) + C\), where \(C\) is the constant of integration.
3. Since the argument of the secant squared function is \(\frac{s}{10}\) instead of \(s\), use substitution to handle the inner function. Let \(u = \frac{s}{10}\).
4. Differentiate \(u\) with respect to \(s\) to find \(du\): \(du = \frac{1}{10} ds\), which implies \(ds = 10 \, du\).
5. Rewrite the integral in terms of \(u\): \(\int \sec^{2}(u) \, (10 \, du) = 10 \int \sec^{2}(u) \, du\). Then integrate using the basic formula to get \(10 \tan(u) + C\). Finally, substitute back \(u = \frac{s}{10}\) to express the answer in terms of \(s\).

Key Concepts

Indefinite Integral

An indefinite integral represents the most general antiderivative of a function, expressed with a constant of integration (C). It reverses differentiation and provides a family of functions whose derivative is the integrand.

Integration of Trigonometric Functions

Certain trigonometric functions have standard integrals, such as ∫sec²(x) dx = tan(x) + C. Recognizing these forms helps in directly integrating expressions involving trigonometric functions.

Substitution Method

When the integrand involves a function inside another function, substitution simplifies the integral by changing variables. For example, setting u = s/10 transforms the integral into a standard form easier to solve.