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measgachyx5q9 2022-05-06 Answered
What is 8 9  x cos ( 3 x ) d x?
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Answers (1)

Ellie Meyers
Answered 2022-05-07 Author has 15 answers

Simplify.

8xcos(3x)9dx

Since 89 is constant with respect to x, move 89 out of the integral.

89xcos(3x)dx

Integrate by parts using the formula udv=uv-vdu, where u=x and dv=cos(3x).

89(x(13sin(3x))-13sin(3x)dx)

Simplify.

89(xsin(3x)3-sin(3x)3dx)

Since 13 is constant with respect to x, move 13 out of the integral.

89(xsin(3x)3-(13sin(3x)dx))

Let u=3x. Then du=3dx, so 13du=dx. Rewrite using u and du.

89(xsin(3x)3-13sin(u)13du)

Combine sin(u) and 13.

89(xsin(3x)3-13sin(u)3du)

Since 13 is constant with respect to u, move 13 out of the integral.

89(xsin(3x)3-13(13sin(u)du))

Simplify.

89(xsin(3x)3-19sin(u)du)

The integral of sin(u) with respect to u is -cos(u).

89(xsin(3x)3-19(-cos(u)+C))

Rewrite 89(xsin(3x)3-19(-cos(u)+C)) as 89(xsin(3x)3+cos(u)9)+C.

89(xsin(3x)3+cos(u)9)+C

Replace all occurrences of u with 3x.

89(xsin(3x)3+cos(3x)9)+C

Simplify.

8xsin(3x)27+8cos(3x)81+C

Reorder terms.

827xsin(3x)+881cos(3x)+C

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