Evaluate the integrals in Exercises 1–22.
∫ 7cos⁷(t) dt

Verified step by step guidance

1. Recognize that the integral involves a power of cosine: \(\int 7 \cos^{7}(t) \, dt\). The constant 7 can be factored out, so rewrite the integral as \(7 \int \cos^{7}(t) \, dt\).
2. Express \(\cos^{7}(t)\) as \(\cos^{6}(t) \cos(t)\) to prepare for substitution: \(7 \int \cos^{6}(t) \cos(t) \, dt\).
3. Rewrite \(\cos^{6}(t)\) as \((\cos^{2}(t))^{3}\), and use the Pythagorean identity \(\cos^{2}(t) = 1 - \sin^{2}(t)\) to get \(7 \int (1 - \sin^{2}(t))^{3} \cos(t) \, dt\).
4. Use the substitution \(u = \sin(t)\), which implies \(du = \cos(t) \, dt\). This transforms the integral into \(7 \int (1 - u^{2})^{3} \, du\).
5. Expand the binomial \((1 - u^{2})^{3}\), then integrate the resulting polynomial term-by-term with respect to \(u\). After integration, substitute back \(u = \sin(t)\) to express the answer in terms of \(t\).

Key Concepts

Integration of Trigonometric Functions

This involves techniques to integrate powers of sine and cosine functions. Recognizing patterns or using identities helps simplify the integral before applying standard methods.

Trigonometric Identities

Identities like the power-reduction or Pythagorean identities transform powers of cosine into expressions involving lower powers or sines, making the integral easier to solve.

Substitution Method

Substitution replaces a complex expression with a simpler variable, often used after applying identities to reduce the integral to a basic form that can be integrated directly.