Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(−3csc²x)dx

Verified step by step guidance

1. Recognize that the integral is of the form \(\int -3 \csc^{2}x \, dx\), where \(-3\) is a constant multiplier and \(\csc^{2}x\) is a standard trigonometric function whose integral is known.
2. Recall the basic integral formula: \(\int \csc^{2}x \, dx = -\cot x + C\), where \(C\) is the constant of integration.
3. Use the constant multiple rule for integrals, which allows you to factor out constants: \(\int -3 \csc^{2}x \, dx = -3 \int \csc^{2}x \, dx\).
4. Substitute the known integral result into the expression: \(-3 \int \csc^{2}x \, dx = -3 (-\cot x + C) = 3 \cot x + C'\), where \(C'\) is a new constant of integration.
5. Verify your result by differentiating \(3 \cot x + C'\) and confirming that it equals the original integrand \(-3 \csc^{2}x\).

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 includes all possible functions whose derivative matches the integrand.

Integration of Trigonometric Functions

Integrating trigonometric functions like csc²x involves knowing standard integral formulas, such as ∫csc²x dx = -cot x + C. Recognizing these forms simplifies finding antiderivatives.

Verification by Differentiation

After finding an indefinite integral, differentiating the result should return the original integrand. This step confirms the correctness of the antiderivative and helps identify any errors.