7–84. Evaluate the following integrals.
19. ∫ from 0 to π/2 [sin⁷x] dx

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Recognize that the integral involves a power of sine, specifically sin⁷(x). To simplify, use the reduction formula for powers of sine: ∫sinⁿ(x) dx = (1/n)∫sinⁿ⁻¹(x)cos(x) dx.

Since the integral is definite, with limits from 0 to π/2, consider using a substitution to simplify the integral. Let u = cos(x), which implies du = -sin(x) dx.

Transform the integral using the substitution u = cos(x). The limits of integration change accordingly: when x = 0, u = cos(0) = 1; when x = π/2, u = cos(π/2) = 0.

Rewrite the integral in terms of u: ∫sin⁷(x) dx becomes ∫(-u⁷) du, with the limits of integration now from 1 to 0. The negative sign can be factored out to reverse the limits.

Evaluate the integral ∫u⁷ du using the power rule for integration: ∫uⁿ du = (uⁿ⁺¹)/(n+1). Apply the new limits of integration (from 1 to 0) to find the result.

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Definite Integral

A definite integral calculates the accumulation of a function's values over a specific interval, represented as ∫ from a to b f(x) dx. It provides the net area under the curve of the function f(x) between the limits a and b. In this case, the integral from 0 to π/2 of sin⁷x requires evaluating the area under the curve of sin⁷x within that interval.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate angles to ratios of sides in right triangles. The function sin(x) oscillates between -1 and 1, and its powers, like sin⁷x, affect the shape of the graph. Understanding the behavior of sin(x) is crucial for evaluating integrals involving trigonometric functions, especially when raised to a power.

Integration Techniques

Integration techniques are methods used to evaluate integrals that may not be solvable by basic antiderivatives. Common techniques include substitution, integration by parts, and trigonometric identities. For the integral of sin⁷x, using the identity sin²x = 1 - cos²x can simplify the expression, making it easier to integrate over the specified limits.

7–84. Evaluate the following integrals.19. ∫ from 0 to π/2 [sin⁷x] dx