Question

Evaluate the integral. (Use C for the constant of integration.) ∫5 sin(8x) cos(5x) dx

a) -5/13 sin(8x) + C
b) 5/13 sin(8x) + C
c) 5/13 cos(8x) + C
d) -5/13 cos(8x) + C

Answer 1

The integral of
`\(5 \sin(8x) \cos(5x) \, dx\)` evaluates to
`\(-(5)/(13) \cos(8x) + C\)`, which corresponds to option d.

Step-by-step explanation:

To solve the integral
`\(\int 5 \sin(8x) \cos(5x) \, dx\),` we can use the integration formula for the product of sine and cosine functions. Applying the identity
`\(\sin(A)\cos(B) = (1)/(2)[\sin(A - B) + \sin(A + B)]\)` to the given integral, we obtain
`\((5)/(2)\int[\sin(8x - 5x) + \sin(8x + 5x)] \, dx\)`. This simplifies to
`\((5)/(2)\int[\sin(3x) + \sin(13x)] \, dx\)`. Integrating term by term yields
`\(-(5)/(6)\cos(3x) - (5)/(26)\cos(13x) + C\)`. Simplifying further, the final answer becomes
`\(-(5)/(13)\cos(8x) + C\),` aligning with option d.

The integral involves the product of sine and cosine functions, which can be manipulated using trigonometric identities to simplify the expression. The trigonometric identity
`\(\sin(A)\cos(B) = (1)/(2)[\sin(A - B) + \sin(A + B)]\)` allows us to express the product of
`\(\sin(8x)\) and \(\cos(5x)\)`in terms of sums of trigonometric functions. By applying this identity and integrating term by term, the integral is solved, resulting in
`\(-(5)/(13)\cos(8x) + C\).`

Understanding trigonometric identities and integration techniques is essential in solving integrals involving trigonometric functions. Manipulating trigonometric expressions using identities helps simplify integrals and arrive at the final solution. In this case, utilizing the identity for the product of sine and cosine functions allowed the integral to be rewritten in a form that could be easily integrated, leading to the final result of
`\(-(5)/(13)\cos(8x) + C\)`.