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Iyana Macdonald

Iyana Macdonald

Answered question

2022-05-06

What is 3 4  t cos ( 5 t ) d t?

Answer & Explanation

Chloe Melendez

Chloe Melendez

Beginner2022-05-07Added 12 answers

Simplify.

3tcos(5t)4dt

Since 34 is constant with respect to t, move 34 out of the integral.

34tcos(5t)dt

Integrate by parts using the formula udv=uv-vdu, where u=t and dv=cos(5t).

34(t(15sin(5t))-15sin(5t)dt)

Simplify.

34(tsin(5t)5-sin(5t)5dt)

Since 15 is constant with respect to t, move 15 out of the integral.

34(tsin(5t)5-(15sin(5t)dt))

Let u=5t. Then du=5dt, so 15du=dt. Rewrite using u and du.

34(tsin(5t)5-15sin(u)15du)

Combine sin(u) and 15.

34(tsin(5t)5-15sin(u)5du)

Since 15 is constant with respect to u, move 15 out of the integral.

34(tsin(5t)5-15(15sin(u)du))

Simplify.

34(tsin(5t)5-125sin(u)du)

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

34(tsin(5t)5-125(-cos(u)+C))

Rewrite 34(tsin(5t)5-125(-cos(u)+C)) as 34(tsin(5t)5+cos(u)25)+C.

34(tsin(5t)5+cos(u)25)+C

Replace all occurrences of u with 5t.

34(tsin(5t)5+cos(5t)25)+C

Reorder terms.

34(15tsin(5t)+125cos(5t))+C

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