Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
(-3/2)csc²x(3x/2)

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Identify the function to find the antiderivative of: $-\frac{3}{2} \csc^{2}x \left( \frac{3x}{2} \right)$.

Rewrite the function for clarity: $-\frac{3}{2} \cdot \frac{3x}{2} \cdot \csc^{2}x = -\frac{9x}{4} \csc^{2}x$.

Recognize that the function is a product of $x$ and $\csc^{2}x$, so consider using integration by parts, where you let $u = x$ and $dv = \csc^{2}x \, dx$.

Compute $du = dx$ and find $v$ by integrating $dv$: since $\int \csc^{2}x \, dx = -\cot x$, then $v = -\cot x$.

Apply the integration by parts formula: $\int u \, dv = uv - \int v \, du$, so the antiderivative is $-\frac{9}{4} \left( x(-\cot x) - \int (-\cot x) \, dx \right)$.

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Antiderivatives (Indefinite Integrals)

An antiderivative of a function is another function whose derivative equals the original function. Finding antiderivatives involves reversing differentiation, often using known integral formulas. The result includes an arbitrary constant since differentiation of a constant is zero.

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Introduction to Indefinite Integrals

Trigonometric Functions and Their Derivatives

Understanding the derivatives of trigonometric functions like csc²x is essential. For example, the derivative of cotangent is -csc²x, which helps identify antiderivatives involving csc²x. Recognizing these relationships simplifies integration of trigonometric expressions.

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Chain Rule and Substitution Method

When functions are composed, such as csc²(3x/2), the chain rule applies in differentiation. To find antiderivatives, substitution reverses this process by setting the inner function as a variable, simplifying the integral and allowing use of basic integral formulas.

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