Use any method to evaluate the integrals in Exercises 65–70.
∫ x cos³(x) dx

Verified step by step guidance

1. Recognize that the integral involves a product of \(x\) and \(\cos^3(x)\), which suggests using integration by parts since \(x\) is a polynomial and \(\cos^3(x)\) is a trigonometric function.
2. Set up integration by parts with \(u = x\) (so that \(du = dx\)) and \(dv = \cos^3(x) \, dx\). The integral then becomes \(\int x \cos^3(x) \, dx = u v - \int v \, du\).
3. To find \(v\), evaluate \(\int \cos^3(x) \, dx\). Use the trigonometric identity \(\cos^3(x) = \cos(x) \cdot \cos^2(x) = \cos(x)(1 - \sin^2(x))\) to rewrite the integral as \(\int \cos(x) (1 - \sin^2(x)) \, dx\).
4. Make the substitution \(t = \sin(x)\), so \(dt = \cos(x) \, dx\). This transforms the integral into \(\int (1 - t^2) \, dt\), which is straightforward to integrate.
5. After finding \(v\), substitute back into the integration by parts formula and simplify the resulting integral \(\int v \, du\) to complete the evaluation.

Key Concepts

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely simplifies the integral, especially when one function becomes simpler upon differentiation.

Trigonometric Identities

Trigonometric identities, such as expressing powers of cosine in terms of multiple angles or lower powers, help simplify integrals involving trigonometric functions. For example, cos³(x) can be rewritten using the identity cos³(x) = cos(x)(1 - sin²(x)) or using power-reduction formulas to facilitate integration.

Substitution Method

The substitution method involves changing variables to simplify an integral. By letting a part of the integrand equal a new variable, the integral can be rewritten in a simpler form. This is especially useful when the integral contains composite functions or when combined with integration by parts.