Find the integral of

sg101cp6vv

sg101cp6vv

Answered question

2022-04-04

Find the integral of 2 5  t sin ( 3 t ) with respect to t.

Answer & Explanation

Haylie Cherry

Haylie Cherry

Beginner2022-04-05Added 17 answers

Simplify.

2tsin(3t)5dt

Since 25 is constant with respect to t, move 25 out of the integral.

25tsin(3t)dt

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

25(t(-13cos(3t))--13cos(3t)dt)

Simplify.

25(-tcos(3t)3--cos(3t)3dt)

Since -1 is constant with respect to t, move -1 out of the integral.

25(-tcos(3t)3--cos(3t)3dt)

Simplify.

25(-tcos(3t)3+cos(3t)3dt)

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

25(-tcos(3t)3+13cos(3t)dt)

Let u=3t. Then du=3dt, so 13du=dt. Rewrite using u and du.

25(-tcos(3t)3+13cos(u)13du)

Combine cos(u) and 13.

25(-tcos(3t)3+13cos(u)3du)

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

25(-tcos(3t)3+13(13cos(u)du))

Simplify.

25(-tcos(3t)3+19cos(u)du)

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

25(-tcos(3t)3+19(sin(u)+C))

Simplify.

25(-cos(3t)t3+sin(u)9)+C

Replace all occurrences of u with 3t.

25(-cos(3t)t3+sin(3t)9)+C

Simplify.

-2tcos(3t)15+2sin(3t)45+C

Reorder terms.

-215tcos(3t)+245sin(3t)+C

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