Final answer:
To evaluate the integral 17 sin(x) cos(x) dx by four methods, we can use integration by substitution, integration by parts, trigonometric identities, or the double angle formula. Each method involves different techniques to find the antiderivative of the given function.
Step-by-step explanation:
To evaluate the integral 17 sin(x) cos(x) dx by four methods, we can use different techniques such as integration by substitution, integration by parts, trigonometric identities, or the double angle formula.
Method 1: Integration by Substitution
Let u = sin(x), then du = cos(x) dx. Substituting these values, we have:
17 sin(x) cos(x) dx = 17 u du
Integrating, we get:
17 integral(u du) = 17(u^2/2) + C = 17/2 sin^2(x) + C
Method 2: Integration by Parts
Let u = sin(x) and dv = 17 cos(x) dx. Then, du = cos(x) dx and v = 17 sin(x). Applying the integration by parts formula:
17 sin(x) cos(x) dx = u v - integral(v du)
17 sin(x) cos(x) dx = 17 sin(x) sin(x) - integral(17 sin(x) cos(x) dx)
Simplifying and rearranging, we get:
2 integral(17 sin(x) cos(x) dx) = 17 sin^2(x)
Integrating both sides, we have:
2(17/2) integral(sin(x) cos(x) dx) = 17 integral(sin^2(x))
17 integral(sin(x) cos(x) dx) = 17/2 sin^2(x) + C
Method 3: Trigonometric Identities
Using the trigonometric identity sin(2x) = 2 sin(x) cos(x), we can rewrite the integral as:
17 sin(x) cos(x) dx = 1/2 * 17 sin(2x) dx
Integrating, we get:
1/2 * 17 integral(sin(2x) dx) = (17/4) cos(2x) + C
Method 4: Double Angle Formula
Using the double angle formula, sin(2x) = 2 sin(x) cos(x), we can rewrite the integral as:
17 sin(x) cos(x) dx = (17/2) sin(x) (1 - sin^2(x)) dx
Expanding and integrating, we get:
(17/2) integral(sin(x) - sin^3(x)) dx = (-17/2) cos(x) + (17/4) cos^3(x) + C